Final Group Project: Network Analysis For Happy Potter
Group 8: Vikas Agarwal, Camille Blain-Coallier, Federico C. Loguercio, Giulio De Felice, Nayla Fakhoury, Alejandro Koury, Victor Vu Duy Phuoc
15/03/2019
Final Project Goal
Harry Potter’s Network:
For our final project, we chose to analyze Harry Potter’s Network. As a new Harry Potter game is coming out on the Iphone/Android (similar to Pokemon Go), we decided to help people make their character pick easier.
Therefore, we chose to analyze this network to evaluate if Harry Potter, the protagonist and supposedly main character of the movies, is actually the most important character within the story line (and the best choice for the game). We will base our conclusion by exploring the following network through various network techniques and subjects:
- Network Properties
- Network Measures: Centrality
- Network Measures: Transitivity
- Network Models: Real World Networks
- Information Diffusion
- Community Detection
- Network Visualization
In addition, we wanted to evaluate which other characters played important roles in the story line.
Clearing and Loading Libraries
Load the mandatory libraries to run the whole project:
- data.table: Provides an enhanced version of data.frames, which are the standard data structure for storing data in base R.
- igraph: Routines for simple graphs and network analysis. It can handle large graphs very well and provides functions for generating random and regular graphs, graph visualization, centrality methods and much more.
- visNetwork: Provides an R interface to the ‘vis.js’ JavaScript charting library. It allows an interactive visualization of networks.
- CINNA: Functions for computing, comparing and demonstrating top informative centrality measures within a network.
- centiserve: Calculates centrality indices additional to the ‘igraph’ package centrality functions.
- keyplayer: Computes group centrality scores and identifies the most central group of players in a network.
- dplyr: A fast, consistent tool for working with data frame like objects, both in memory and out of memory.
- sna: A range of tools for social network analysis, including node and graph-level indices, structural distance and covariance methods, structural equivalence detection, network regression, random graph generation, and 2D/3D network visualization.
#dev.off() #should be run only if R is not under default parameters
rm(list=ls()) #clear environment from past data
list_of_packages <-c("data.table", "igraph", "visNetwork", "CINNA", "centiserve", "keyplayer", "dplyr", "sna")
new_packages <- list_of_packages[!(list_of_packages %in% installed.packages()[,"Package"])]
if(length(new_packages)) install.packages(new_packages) #install libraries from the list of packages only if they haven't already been installed
library("data.table")
library("igraph")## Warning: package 'igraph' was built under R version 3.5.2##
## Attaching package: 'igraph'## The following objects are masked from 'package:stats':
##
## decompose, spectrum## The following object is masked from 'package:base':
##
## unionlibrary("CINNA")## Warning: package 'CINNA' was built under R version 3.5.2library("visNetwork")## Warning: package 'visNetwork' was built under R version 3.5.2library("centiserve")## Warning: package 'centiserve' was built under R version 3.5.2## Loading required package: Matrixlibrary("keyplayer")## Warning: package 'keyplayer' was built under R version 3.5.2##
## Attaching package: 'keyplayer'## The following object is masked from 'package:igraph':
##
## contractlibrary("dplyr")## Warning: package 'dplyr' was built under R version 3.5.2##
## Attaching package: 'dplyr'## The following objects are masked from 'package:igraph':
##
## as_data_frame, groups, union## The following objects are masked from 'package:data.table':
##
## between, first, last## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary("sna")## Warning: package 'sna' was built under R version 3.5.2## Loading required package: statnet.common## Warning: package 'statnet.common' was built under R version 3.5.2##
## Attaching package: 'statnet.common'## The following object is masked from 'package:base':
##
## order## Loading required package: network## Warning: package 'network' was built under R version 3.5.2## network: Classes for Relational Data
## Version 1.13.0.1 created on 2015-08-31.
## copyright (c) 2005, Carter T. Butts, University of California-Irvine
## Mark S. Handcock, University of California -- Los Angeles
## David R. Hunter, Penn State University
## Martina Morris, University of Washington
## Skye Bender-deMoll, University of Washington
## For citation information, type citation("network").
## Type help("network-package") to get started.##
## Attaching package: 'network'## The following objects are masked from 'package:igraph':
##
## %c%, %s%, add.edges, add.vertices, delete.edges,
## delete.vertices, get.edge.attribute, get.edges,
## get.vertex.attribute, is.bipartite, is.directed,
## list.edge.attributes, list.vertex.attributes,
## set.edge.attribute, set.vertex.attribute## sna: Tools for Social Network Analysis
## Version 2.4 created on 2016-07-23.
## copyright (c) 2005, Carter T. Butts, University of California-Irvine
## For citation information, type citation("sna").
## Type help(package="sna") to get started.##
## Attaching package: 'sna'## The following objects are masked from 'package:igraph':
##
## betweenness, bonpow, closeness, components, degree,
## dyad.census, evcent, hierarchy, is.connected, neighborhood,
## triad.census#load all libraries from the list of packagesLoading Data
In this section, the goal is loading the given datasets, building the graph and analyzing basics metrics.
Harry Potter’s Network:
There are two data frames that were loaded, the Edges and the Keys.
The Keys dataframe shows a list of 65 Characters and includes information regarding their alignment, gender, house to which they belong and the color attributed to that house. These 65 characters are regarded as the most important characters within all the Harry Potter movies.
The Edge dataframe shows 513 links between characters (links between each character name (source) with another character (target)). However, many have duplicated loops, so that when we remove those, the number of edges decreases to 330. There is no need for duplicated loops in this case because our graph is undirected and if one character knows another, then automatically, the inverse is true.
In all, the joint graph (igraph) of the edges and keys is an undirected and unweighted network.The dataframe is unweighted because the links are based on which Characters know each other.
edges <- as.data.frame(fread("https://gist.githubusercontent.com/CamilleBlainCoallier/3e5604911cad5004bfe213030291808a/raw/58bdd7a0dce34e0f5de309fe8d6c1ce536ef148b/Edge_list_HP"))
keys <- as.data.frame(fread("https://gist.githubusercontent.com/CamilleBlainCoallier/7c564144a0c8397fac1b6b67c5776616/raw/2be29fbc24fcb55022b07ce40cc4d74786821e61/Keys_nodes_HP"))
summary(edges) #list of links between characters (target to source)## Source Target
## NULL:Regulus Arcturus Black NULL:Sirius Black
## NULL:Regulus Arcturus Black NULL:Bellatrix Lestrange
## NULL:Regulus Arcturus Black NULL:Lord Voldemort
## NULL:Sirius Black NULL:Regulus Arcturus Black
## NULL:Sirius Black NULL:Albus Dumbledore
## NULL:Sirius Black NULL:Hermione Granger
## NULL:Sirius Black NULL:Bellatrix Lestrange
## NULL:Sirius Black NULL:Remus Lupin
## NULL:Sirius Black NULL:Lucius Malfoy
## NULL:Sirius Black NULL:Narcissa Malfoy
## NULL:Sirius Black NULL:Minerva McGonagall
## NULL:Sirius Black NULL:Alastor Moody
## NULL:Sirius Black NULL:Peter Pettigrew
## NULL:Sirius Black NULL:Harry Potter
## NULL:Sirius Black NULL:James Potter
## NULL:Sirius Black NULL:Lily Potter
## NULL:Sirius Black NULL:Lord Voldemort
## NULL:Sirius Black NULL:Severus Snape
## NULL:Sirius Black NULL:Nymphadora Tonks
## NULL:Sirius Black NULL:Arthur Weasley
## NULL:Sirius Black NULL:Fred Weasley
## NULL:Sirius Black NULL:George Weasley
## NULL:Sirius Black NULL:Ginny Weasley
## NULL:Sirius Black NULL:Ron Weasley
## NULL:Lavender Brown NULL:Hermione Granger
## NULL:Lavender Brown NULL:Neville Longbottom
## NULL:Lavender Brown NULL:Ron Weasley
## NULL:Cho Chang NULL:Cedric Diggory
## NULL:Cho Chang NULL:Seamus Finnigan
## NULL:Cho Chang NULL:Hermione Granger
## NULL:Cho Chang NULL:Neville Longbottom
## NULL:Cho Chang NULL:Harry Potter
## NULL:Cho Chang NULL:Lord Voldemort
## NULL:Cho Chang NULL:Ron Weasley
## NULL:Vincent Crabbe Sr. NULL:Regulus Arcturus Black
## NULL:Vincent Crabbe Sr. NULL:Vincent Crabbe
## NULL:Vincent Crabbe Sr. NULL:Bartemius Crouch Sr.
## NULL:Vincent Crabbe Sr. NULL:Bellatrix Lestrange
## NULL:Vincent Crabbe Sr. NULL:Lucius Malfoy
## NULL:Vincent Crabbe Sr. NULL:Peter Pettigrew
## NULL:Vincent Crabbe Sr. NULL:Lord Voldemort
## NULL:Vincent Crabbe Sr. NULL:Severus Snape
## NULL:Vincent Crabbe NULL:Vincent Crabbe Sr.
## NULL:Vincent Crabbe NULL:Dolores Janes Umbridge
## NULL:Vincent Crabbe NULL:Gregory Goyle
## NULL:Vincent Crabbe NULL:Hermione Granger
## NULL:Vincent Crabbe NULL:Draco Malfoy
## NULL:Vincent Crabbe NULL:Harry Potter
## NULL:Vincent Crabbe NULL:Ron Weasley
## NULL:Bartemius Crouch Sr. NULL:Vincent Crabbe Sr.
## NULL:Bartemius Crouch Sr. NULL:Bartemius Crouch Jr.
## NULL:Bartemius Crouch Sr. NULL:Igor Karkaroff
## NULL:Bartemius Crouch Sr. NULL:Percy Weasley
## NULL:Bartemius Crouch Jr. NULL:Bartemius Crouch Sr.
## NULL:Bartemius Crouch Jr. NULL:Hermione Granger
## NULL:Bartemius Crouch Jr. NULL:Neville Longbottom
## NULL:Bartemius Crouch Jr. NULL:Minerva McGonagall
## NULL:Bartemius Crouch Jr. NULL:Alastor Moody
## NULL:Bartemius Crouch Jr. NULL:Harry Potter
## NULL:Bartemius Crouch Jr. NULL:Lord Voldemort
## NULL:Bartemius Crouch Jr. NULL:Severus Snape
## NULL:Bartemius Crouch Jr. NULL:Peter Pettigrew
## NULL:Fleur Delacour NULL:Hermione Granger
## NULL:Fleur Delacour NULL:Bellatrix Lestrange
## NULL:Fleur Delacour NULL:Neville Longbottom
## NULL:Fleur Delacour NULL:Alastor Moody
## NULL:Fleur Delacour NULL:Harry Potter
## NULL:Fleur Delacour NULL:Bill Weasley
## NULL:Fleur Delacour NULL:Ginny Weasley
## NULL:Fleur Delacour NULL:Molly Weasley
## NULL:Fleur Delacour NULL:Ron Weasley
## NULL:Cedric Diggory NULL:Cho Chang
## NULL:Cedric Diggory NULL:Harry Potter
## NULL:Cedric Diggory NULL:Lord Voldemort
## NULL:Aberforth Dumbledore NULL:Albus Dumbledore
## NULL:Aberforth Dumbledore NULL:Hermione Granger
## NULL:Aberforth Dumbledore NULL:Harry Potter
## NULL:Aberforth Dumbledore NULL:Ron Weasley
## NULL:Aberforth Dumbledore NULL:Dobby
## NULL:Albus Dumbledore NULL:Sirius Black
## NULL:Albus Dumbledore NULL:Aberforth Dumbledore
## NULL:Albus Dumbledore NULL:Petunia Dursley
## NULL:Albus Dumbledore NULL:Vernon Dursley
## NULL:Albus Dumbledore NULL:Hermione Granger
## NULL:Albus Dumbledore NULL:Rubeus Hagrid
## NULL:Albus Dumbledore NULL:Bellatrix Lestrange
## NULL:Albus Dumbledore NULL:Remus Lupin
## NULL:Albus Dumbledore NULL:Minerva McGonagall
## NULL:Albus Dumbledore NULL:Minerva McGonagall
## NULL:Albus Dumbledore NULL:Alastor Moody
## NULL:Albus Dumbledore NULL:Peter Pettigrew
## NULL:Albus Dumbledore NULL:Harry Potter
## NULL:Albus Dumbledore NULL:James Potter
## NULL:Albus Dumbledore NULL:Lily Potter
## NULL:Albus Dumbledore NULL:Quirinus Quirrell
## NULL:Albus Dumbledore NULL:Lord Voldemort
## NULL:Albus Dumbledore NULL:Rita Skeeter
## NULL:Albus Dumbledore NULL:Severus Snape
## NULL:Albus Dumbledore NULL:Dolores Janes Umbridge
## NULL:Albus Dumbledore NULL:Dolores Janes Umbridge
## NULL:Albus Dumbledore NULL:Arthur Weasley
## NULL:Albus Dumbledore NULL:Bill Weasley
## NULL:Albus Dumbledore NULL:Charlie Weasley
## NULL:Albus Dumbledore NULL:Fred Weasley
## NULL:Albus Dumbledore NULL:George Weasley
## NULL:Albus Dumbledore NULL:Ginny Weasley
## NULL:Albus Dumbledore NULL:Molly Weasley
## NULL:Albus Dumbledore NULL:Percy Weasley
## NULL:Albus Dumbledore NULL:Ron Weasley
## NULL:Dudley Dursley NULL:Dudley Dursley
## NULL:Dudley Dursley NULL:Vernon Dursley
## NULL:Dudley Dursley NULL:Harry Potter
## NULL:Dudley Dursley NULL:Rubeus Hagrid
## NULL:Petunia Dursley NULL:Albus Dumbledore
## NULL:Petunia Dursley NULL:Dudley Dursley
## NULL:Petunia Dursley NULL:Petunia Dursley
## NULL:Petunia Dursley NULL:Harry Potter
## NULL:Petunia Dursley NULL:Lily Potter
## NULL:Vernon Dursley NULL:Albus Dumbledore
## NULL:Vernon Dursley NULL:Dudley Dursley
## NULL:Vernon Dursley NULL:Petunia Dursley
## NULL:Vernon Dursley NULL:Harry Potter
## NULL:Argus Filch NULL:Sirius Black
## NULL:Argus Filch NULL:Albus Dumbledore
## NULL:Argus Filch NULL:Rubeus Hagrid
## NULL:Argus Filch NULL:Minerva McGonagall
## NULL:Argus Filch NULL:Peter Pettigrew
## NULL:Argus Filch NULL:Harry Potter
## NULL:Argus Filch NULL:James Potter
## NULL:Argus Filch NULL:Severus Snape
## NULL:Argus Filch NULL:Dolores Janes Umbridge
## NULL:Argus Filch NULL:Fred Weasley
## NULL:Argus Filch NULL:George Weasley
## NULL:Seamus Finnigan NULL:Lavender Brown
## NULL:Seamus Finnigan NULL:Cho Chang
## NULL:Seamus Finnigan NULL:Hermione Granger
## NULL:Seamus Finnigan NULL:Neville Longbottom
## NULL:Seamus Finnigan NULL:Luna Lovegood
## NULL:Seamus Finnigan NULL:Harry Potter
## NULL:Seamus Finnigan NULL:Ginny Weasley
## NULL:Seamus Finnigan NULL:Ron Weasley
## NULL:Nicolas Flamel NULL:Albus Dumbledore
## NULL:Cornelius Fudge NULL:Albus Dumbledore
## NULL:Cornelius Fudge NULL:Lucius Malfoy
## NULL:Cornelius Fudge NULL:Harry Potter
## NULL:Cornelius Fudge NULL:Dolores Janes Umbridge
## NULL:Cornelius Fudge NULL:Arthur Weasley
## NULL:Cornelius Fudge NULL:Percy Weasley
## NULL:Goyle Sr. NULL:Vincent Crabbe Sr.
## NULL:Goyle Sr. NULL:Gregory Goyle
## NULL:Goyle Sr. NULL:Lord Voldemort
## NULL:Gregory Goyle NULL:Goyle Sr.
## NULL:Gregory Goyle NULL:Vincent Crabbe
## NULL:Gregory Goyle NULL:Draco Malfoy
## NULL:Gregory Goyle NULL:Harry Potter
## NULL:Gregory Goyle NULL:Lord Voldemort
## NULL:Hermione Granger NULL:Harry Potter
## NULL:Hermione Granger NULL:Ron Weasley
## NULL:Hermione Granger NULL:George Weasley
## NULL:Hermione Granger NULL:Molly Weasley
## NULL:Hermione Granger NULL:Severus Snape
## NULL:Hermione Granger NULL:Luna Lovegood
## NULL:Hermione Granger NULL:Neville Longbottom
## NULL:Hermione Granger NULL:Lord Voldemort
## NULL:Hermione Granger NULL:Peter Pettigrew
## NULL:Hermione Granger NULL:Draco Malfoy
## NULL:Hermione Granger NULL:Argus Filch
## NULL:Hermione Granger NULL:Gregory Goyle
## NULL:Rubeus Hagrid NULL:Lord Voldemort
## NULL:Rubeus Hagrid NULL:Fluffy
## NULL:Rubeus Hagrid NULL:Aragog
## NULL:Rubeus Hagrid NULL:Grawp
## NULL:Rubeus Hagrid NULL:Ron Weasley
## NULL:Rubeus Hagrid NULL:Hermione Granger
## NULL:Rubeus Hagrid NULL:Severus Snape
## NULL:Rubeus Hagrid NULL:Harry Potter
## NULL:Rubeus Hagrid NULL:Neville Longbottom
## NULL:Rubeus Hagrid NULL:Olympe Maxime
## NULL:Rubeus Hagrid NULL:Minerva McGonagall
## NULL:Rubeus Hagrid NULL:Albus Dumbledore
## NULL:Igor Karkaroff NULL:Lord Voldemort
## NULL:Igor Karkaroff NULL:Viktor Krum
## NULL:Viktor Krum NULL:Igor Karkaroff
## NULL:Viktor Krum NULL:Harry Potter
## NULL:Viktor Krum NULL:Fleur Delacour
## NULL:Viktor Krum NULL:Cedric Diggory
## NULL:Viktor Krum NULL:Hermione Granger
## NULL:Bellatrix Lestrange NULL:Lord Voldemort
## NULL:Bellatrix Lestrange NULL:Lucius Malfoy
## NULL:Bellatrix Lestrange NULL:Narcissa Malfoy
## NULL:Bellatrix Lestrange NULL:Sirius Black
## NULL:Bellatrix Lestrange NULL:Harry Potter
## NULL:Bellatrix Lestrange NULL:Nymphadora Tonks
## NULL:Bellatrix Lestrange NULL:Molly Weasley
## NULL:Bellatrix Lestrange NULL:Dobby
## NULL:Alice Longbottom NULL:Frank Longbottom
## NULL:Alice Longbottom NULL:Frank Longbottom
## NULL:Alice Longbottom NULL:Neville Longbottom
## NULL:Frank Longbottom NULL:Alice Longbottom
## NULL:Frank Longbottom NULL:Lord Voldemort
## NULL:Frank Longbottom NULL:Neville Longbottom
## NULL:Neville Longbottom NULL:Harry Potter
## NULL:Neville Longbottom NULL:Lord Voldemort
## NULL:Neville Longbottom NULL:Hermione Granger
## NULL:Neville Longbottom NULL:Ron Weasley
## NULL:Neville Longbottom NULL:Luna Lovegood
## NULL:Neville Longbottom NULL:Draco Malfoy
## NULL:Luna Lovegood NULL:Neville Longbottom
## NULL:Luna Lovegood NULL:Harry Potter
## NULL:Luna Lovegood NULL:Hermione Granger
## NULL:Luna Lovegood NULL:Ron Weasley
## NULL:Xenophilius Lovegood NULL:Luna Lovegood
## NULL:Remus Lupin NULL:Peter Pettigrew
## NULL:Remus Lupin NULL:Lord Voldemort
## NULL:Remus Lupin NULL:James Potter
## NULL:Remus Lupin NULL:Sirius Black
## NULL:Remus Lupin NULL:Nymphadora Tonks
## NULL:Remus Lupin NULL:Harry Potter
## NULL:Remus Lupin NULL:Minerva McGonagall
## NULL:Draco Malfoy NULL:Vincent Crabbe
## NULL:Draco Malfoy NULL:Gregory Goyle
## NULL:Draco Malfoy NULL:Lucius Malfoy
## NULL:Draco Malfoy NULL:Narcissa Malfoy
## NULL:Draco Malfoy NULL:Harry Potter
## NULL:Draco Malfoy NULL:Hermione Granger
## NULL:Draco Malfoy NULL:Ron Weasley
## NULL:Lucius Malfoy NULL:Draco Malfoy
## NULL:Lucius Malfoy NULL:Narcissa Malfoy
## NULL:Lucius Malfoy NULL:Lord Voldemort
## NULL:Lucius Malfoy NULL:Harry Potter
## NULL:Lucius Malfoy NULL:Dobby
## NULL:Narcissa Malfoy NULL:Lucius Malfoy
## NULL:Narcissa Malfoy NULL:Lord Voldemort
## NULL:Narcissa Malfoy NULL:Bellatrix Lestrange
## NULL:Olympe Maxime NULL:Rubeus Hagrid
## NULL:Olympe Maxime NULL:Fleur Delacour
## NULL:Olympe Maxime NULL:Albus Dumbledore
## NULL:Olympe Maxime NULL:Igor Karkaroff
## NULL:Minerva McGonagall NULL:Argus Filch
## NULL:Minerva McGonagall NULL:Rubeus Hagrid
## NULL:Minerva McGonagall NULL:Dolores Janes Umbridge
## NULL:Minerva McGonagall NULL:Lord Voldemort
## NULL:Minerva McGonagall NULL:Albus Dumbledore
## NULL:Minerva McGonagall NULL:Harry Potter
## NULL:Harry Potter NULL:Minerva McGonagall
## NULL:Minerva McGonagall NULL:Hermione Granger
## NULL:Hermione Granger NULL:Minerva McGonagall
## NULL:Minerva McGonagall NULL:Ron Weasley
## NULL:Ron Weasley NULL:Minerva McGonagall
## NULL:Minerva McGonagall NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:Minerva McGonagall
## NULL:Minerva McGonagall NULL:Albus Dumbledore
## NULL:Albus Dumbledore NULL:Minerva McGonagall
## NULL:Alastor Moody NULL:Harry Potter
## NULL:Alastor Moody NULL:Arthur Weasley
## NULL:Alastor Moody NULL:Molly Weasley
## NULL:Harry Potter NULL:Alastor Moody
## NULL:Harry Potter NULL:Lord Voldemort
## NULL:Alastor Moody NULL:Hermione Granger
## NULL:Hermione Granger NULL:Alastor Moody
## NULL:Alastor Moody NULL:Hermione Granger
## NULL:Hermione Granger NULL:Alastor Moody
## NULL:Alastor Moody NULL:Bartemius Crouch Jr.
## NULL:Bartemius Crouch Jr. NULL:Alastor Moody
## NULL:Alastor Moody NULL:Albus Dumbledore
## NULL:Albus Dumbledore NULL:Alastor Moody
## NULL:Alastor Moody NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:Alastor Moody
## NULL:Peter Pettigrew NULL:Sirius Black
## NULL:Sirius Black NULL:Peter Pettigrew
## NULL:Peter Pettigrew NULL:James Potter
## NULL:James Potter NULL:Peter Pettigrew
## NULL:Peter Pettigrew NULL:Lily Potter
## NULL:Lily Potter NULL:Peter Pettigrew
## NULL:Lily Potter NULL:Harry Potter
## NULL:Peter Pettigrew NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:Peter Pettigrew
## NULL:Peter Pettigrew NULL:Harry Potter
## NULL:Harry Potter NULL:Peter Pettigrew
## NULL:James Potter NULL:Harry Potter
## NULL:Harry Potter NULL:James Potter
## NULL:James Potter NULL:Remus Lupin
## NULL:Remus Lupin NULL:James Potter
## NULL:James Potter NULL:Sirius Black
## NULL:Sirius Black NULL:James Potter
## NULL:James Potter NULL:Peter Pettigrew
## NULL:Peter Pettigrew NULL:James Potter
## NULL:James Potter NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:James Potter
## NULL:James Potter NULL:Lily Potter
## NULL:Lily Potter NULL:James Potter
## NULL:Lily Potter NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:Lily Potter
## NULL:Lily Potter NULL:Severus Snape
## NULL:Severus Snape NULL:Lily Potter
## NULL:Quirinus Quirrell NULL:Harry Potter
## NULL:Harry Potter NULL:Quirinus Quirrell
## NULL:Quirinus Quirrell NULL:Hermione Granger
## NULL:Hermione Granger NULL:Quirinus Quirrell
## NULL:Quirinus Quirrell NULL:Hermione Granger
## NULL:Hermione Granger NULL:Quirinus Quirrell
## NULL:Quirinus Quirrell NULL:Albus Dumbledore
## NULL:Albus Dumbledore NULL:Quirinus Quirrell
## NULL:Quirinus Quirrell NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:Quirinus Quirrell
## NULL:Tom Riddle Sr. NULL:Lord Voldemort
## NULL:Tom Riddle Sr. NULL:Tom Riddle Sr.
## NULL:Tom Riddle Sr. NULL:Mary Riddle
## NULL:Mary Riddle NULL:Tom Riddle Sr.
## NULL:Mary Riddle NULL:Lord Voldemort
## NULL:Tom Riddle Sr. NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:Tom Riddle Sr.
## NULL:Rita Skeeter NULL:Harry Potter
## NULL:Harry Potter NULL:Rita Skeeter
## NULL:Rita Skeeter NULL:Albus Dumbledore
## NULL:Albus Dumbledore NULL:Rita Skeeter
## NULL:Severus Snape NULL:Harry Potter
## NULL:Harry Potter NULL:Severus Snape
## NULL:Severus Snape NULL:Hermione Granger
## NULL:Hermione Granger NULL:Severus Snape
## NULL:Severus Snape NULL:Hermione Granger
## NULL:Hermione Granger NULL:Severus Snape
## NULL:Quirinus Quirrell NULL:Albus Dumbledore
## NULL:Albus Dumbledore NULL:Quirinus Quirrell
## NULL:Severus Snape NULL:Lily Potter
## NULL:Lily Potter NULL:Severus Snape
## NULL:Severus Snape NULL:Lord Voldemort
## NULL:Lord Voldemort NULL:Severus Snape
## NULL:Nymphadora Tonks NULL:Remus Lupin
## NULL:Nymphadora Tonks NULL:Lord Voldemort
## NULL:Remus Lupin NULL:Nymphadora Tonks
## NULL:Dolores Janes Umbridge NULL:Harry Potter
## NULL:Harry Potter NULL:Dolores Janes Umbridge
## NULL:Dolores Janes Umbridge NULL:Hermione Granger
## NULL:Hermione Granger NULL:Dolores Janes Umbridge
## NULL:Dolores Janes Umbridge NULL:Hermione Granger
## NULL:Hermione Granger NULL:Dolores Janes Umbridge
## NULL:Dolores Janes Umbridge NULL:Albus Dumbledore
## NULL:Albus Dumbledore NULL:Dolores Janes Umbridge
## NULL:Dolores Janes Umbridge NULL:Cornelius Fudge
## NULL:Cornelius Fudge NULL:Dolores Janes Umbridge
## NULL:Arthur Weasley NULL:Hermione Granger
## NULL:Arthur Weasley NULL:Sirius Black
## NULL:Arthur Weasley NULL:Albus Dumbledore
## NULL:Arthur Weasley NULL:Lucius Malfoy
## NULL:Arthur Weasley NULL:Harry Potter
## NULL:Hermione Granger NULL:Arthur Weasley
## NULL:Arthur Weasley NULL:Bill Weasley
## NULL:Bill Weasley NULL:Arthur Weasley
## NULL:Arthur Weasley NULL:Charlie Weasley
## NULL:Charlie Weasley NULL:Arthur Weasley
## NULL:Arthur Weasley NULL:Fred Weasley
## NULL:Fred Weasley NULL:Arthur Weasley
## NULL:Arthur Weasley NULL:George Weasley
## NULL:George Weasley NULL:Arthur Weasley
## NULL:Arthur Weasley NULL:George Weasley
## NULL:George Weasley NULL:Arthur Weasley
## NULL:Arthur Weasley NULL:Ginny Weasley
## NULL:Ginny Weasley NULL:Arthur Weasley
## NULL:Arthur Weasley NULL:Percy Weasley
## NULL:Percy Weasley NULL:Arthur Weasley
## NULL:Bill Weasley NULL:Harry Potter
## NULL:Bill Weasley NULL:Lord Voldemort
## NULL:Bill Weasley NULL:Hermione Granger
## NULL:Hermione Granger NULL:Bill Weasley
## NULL:Arthur Weasley NULL:Bill Weasley
## NULL:Bill Weasley NULL:Arthur Weasley
## NULL:Bill Weasley NULL:Charlie Weasley
## NULL:Charlie Weasley NULL:Bill Weasley
## NULL:Bill Weasley NULL:Fred Weasley
## NULL:Fred Weasley NULL:Bill Weasley
## NULL:Bill Weasley NULL:George Weasley
## NULL:George Weasley NULL:Bill Weasley
## NULL:Bill Weasley NULL:George Weasley
## NULL:George Weasley NULL:Bill Weasley
## NULL:Bill Weasley NULL:Ginny Weasley
## NULL:Ginny Weasley NULL:Bill Weasley
## NULL:Bill Weasley NULL:Percy Weasley
## NULL:Percy Weasley NULL:Bill Weasley
## NULL:Charlie Weasley NULL:Hermione Granger
## NULL:Charlie Weasley NULL:Harry Potter
## NULL:Hermione Granger NULL:Charlie Weasley
## NULL:Arthur Weasley NULL:Charlie Weasley
## NULL:Charlie Weasley NULL:Arthur Weasley
## NULL:Charlie Weasley NULL:Fred Weasley
## NULL:Fred Weasley NULL:Charlie Weasley
## NULL:Charlie Weasley NULL:George Weasley
## NULL:George Weasley NULL:Charlie Weasley
## NULL:Charlie Weasley NULL:George Weasley
## NULL:Charlie Weasley NULL:Lord Voldemort
## NULL:George Weasley NULL:Charlie Weasley
## NULL:Charlie Weasley NULL:Ginny Weasley
## NULL:Ginny Weasley NULL:Charlie Weasley
## NULL:Charlie Weasley NULL:Percy Weasley
## NULL:Percy Weasley NULL:Charlie Weasley
## NULL:Fred Weasley NULL:Hermione Granger
## NULL:Fred Weasley NULL:Remus Lupin
## NULL:Fred Weasley NULL:Peter Pettigrew
## NULL:Fred Weasley NULL:Harry Potter
## NULL:Fred Weasley NULL:Lord Voldemort
## NULL:Fred Weasley NULL:Severus Snape
## NULL:Fred Weasley NULL:Arthur Weasley
## NULL:Fred Weasley NULL:Bill Weasley
## NULL:Fred Weasley NULL:Charlie Weasley
## NULL:Fred Weasley NULL:George Weasley
## NULL:Fred Weasley NULL:Ginny Weasley
## NULL:Fred Weasley NULL:Molly Weasley
## NULL:Fred Weasley NULL:Percy Weasley
## NULL:Fred Weasley NULL:Ron Weasley
## NULL:George Weasley NULL:Hermione Granger
## NULL:George Weasley NULL:Remus Lupin
## NULL:George Weasley NULL:Peter Pettigrew
## NULL:George Weasley NULL:Harry Potter
## NULL:George Weasley NULL:Lord Voldemort
## NULL:George Weasley NULL:Severus Snape
## NULL:George Weasley NULL:Arthur Weasley
## NULL:George Weasley NULL:Bill Weasley
## NULL:George Weasley NULL:Charlie Weasley
## NULL:George Weasley NULL:Fred Weasley
## NULL:George Weasley NULL:Ginny Weasley
## NULL:George Weasley NULL:Molly Weasley
## NULL:George Weasley NULL:Percy Weasley
## NULL:George Weasley NULL:Ron Weasley
## NULL:Ginny Weasley NULL:Hermione Granger
## NULL:Ginny Weasley NULL:Remus Lupin
## NULL:Ginny Weasley NULL:Peter Pettigrew
## NULL:Ginny Weasley NULL:Harry Potter
## NULL:Ginny Weasley NULL:Lord Voldemort
## NULL:Ginny Weasley NULL:Severus Snape
## NULL:Ginny Weasley NULL:Arthur Weasley
## NULL:Ginny Weasley NULL:Bill Weasley
## NULL:Ginny Weasley NULL:Charlie Weasley
## NULL:Ginny Weasley NULL:George Weasley
## NULL:Ginny Weasley NULL:Fred Weasley
## NULL:Ginny Weasley NULL:Molly Weasley
## NULL:Ginny Weasley NULL:Percy Weasley
## NULL:Ginny Weasley NULL:Percy Weasley
## NULL:Ginny Weasley NULL:Ron Weasley
## NULL:Molly Weasley NULL:Sirius Black
## NULL:Molly Weasley NULL:Albus Dumbledore
## NULL:Molly Weasley NULL:Hermione Granger
## NULL:Molly Weasley NULL:Lucius Malfoy
## NULL:Molly Weasley NULL:Harry Potter
## NULL:Molly Weasley NULL:Lord Voldemort
## NULL:Molly Weasley NULL:Arthur Weasley
## NULL:Molly Weasley NULL:Bill Weasley
## NULL:Molly Weasley NULL:Charlie Weasley
## NULL:Molly Weasley NULL:Fred Weasley
## NULL:Molly Weasley NULL:George Weasley
## NULL:Molly Weasley NULL:Ginny Weasley
## NULL:Molly Weasley NULL:Percy Weasley
## NULL:Molly Weasley NULL:Ron Weasley
## NULL:Percy Weasley NULL:Arthur Weasley
## NULL:Percy Weasley NULL:Bill Weasley
## NULL:Percy Weasley NULL:Charlie Weasley
## NULL:Percy Weasley NULL:Fred Weasley
## NULL:Percy Weasley NULL:George Weasley
## NULL:Percy Weasley NULL:Ginny Weasley
## NULL:Percy Weasley NULL:Molly Weasley
## NULL:Percy Weasley NULL:Ron Weasley
## NULL:Ron Weasley NULL:Sirius Black
## NULL:Ron Weasley NULL:Fleur Delacour
## NULL:Ron Weasley NULL:Albus Dumbledore
## NULL:Ron Weasley NULL:Argus Filch
## NULL:Ron Weasley NULL:Seamus Finnigan
## NULL:Ron Weasley NULL:Gregory Goyle
## NULL:Ron Weasley NULL:Hermione Granger
## NULL:Ron Weasley NULL:Rubeus Hagrid
## NULL:Ron Weasley NULL:Bellatrix Lestrange
## NULL:Ron Weasley NULL:Neville Longbottom
## NULL:Ron Weasley NULL:Luna Lovegood
## NULL:Ron Weasley NULL:Remus Lupin
## NULL:Ron Weasley NULL:Draco Malfoy
## NULL:Ron Weasley NULL:Lucius Malfoy
## NULL:Ron Weasley NULL:Minerva McGonagall
## NULL:Ron Weasley NULL:Alastor Moody
## NULL:Ron Weasley NULL:Peter Pettigrew
## NULL:Ron Weasley NULL:Harry Potter
## NULL:Ron Weasley NULL:Lord Voldemort
## NULL:Ron Weasley NULL:Severus Snape
## NULL:Ron Weasley NULL:Dolores Janes Umbridge
## NULL:Ron Weasley NULL:Arthur Weasley
## NULL:Ron Weasley NULL:Bill Weasley
## NULL:Ron Weasley NULL:Charlie Weasley
## NULL:Ron Weasley NULL:Fred Weasley
## NULL:Ron Weasley NULL:George Weasley
## NULL:Ron Weasley NULL:Ginny Weasley
## NULL:Ron Weasley NULL:Molly Weasley
## NULL:Ron Weasley NULL:Percy Weasley
## NULL:Ron Weasley NULL:Dobby
## NULL:Ron Weasley NULL:Fluffy
## NULL:Ron Weasley NULL:Aragog
## NULL:Ron Weasley NULL:Grawp
## NULL:Dobby NULL:Hermione Granger
## NULL:Dobby NULL:Bellatrix Lestrange
## NULL:Dobby NULL:Draco Malfoy
## NULL:Dobby NULL:Lucius Malfoy
## NULL:Dobby NULL:Harry Potter
## NULL:Dobby NULL:Ron Weasley
## NULL:Fluffy NULL:Hermione Granger
## NULL:Fluffy NULL:Rubeus Hagrid
## NULL:Fluffy NULL:Harry Potter
## NULL:Fluffy NULL:Ron Weasley
## NULL:Hedwig NULL:Harry Potter
## NULL:Moaning Myrtle NULL:Harry Potter
## NULL:Aragog NULL:Hermione Granger
## NULL:Aragog NULL:Rubeus Hagrid
## NULL:Aragog NULL:Harry Potter
## NULL:Aragog NULL:Ron Weasley
## NULL:Grawp NULL:Hermione Granger
## NULL:Grawp NULL:Rubeus Hagrid
## NULL:Grawp NULL:Harry Potter
## NULL:Aragog NULL:Ron Weasleyg1<- graph_from_data_frame(d=edges, vertices=keys, directed=F)
g1 #originally, we can see that there are 65 nodes and 513 edges from calling the graph## IGRAPH 861035f UN-- 65 513 --
## + attr: name (v/c), Alignment (v/c), Gender_Male (v/n), House
## | (v/c), Color (v/c)
## + edges from 861035f (vertex names):
## [1] Regulus Arcturus Black--Sirius Black
## [2] Regulus Arcturus Black--Bellatrix Lestrange
## [3] Regulus Arcturus Black--Lord Voldemort
## [4] Regulus Arcturus Black--Sirius Black
## [5] Sirius Black --Albus Dumbledore
## [6] Sirius Black --Hermione Granger
## [7] Sirius Black --Bellatrix Lestrange
## + ... omitted several edgesg11<-igraph::simplify(g1,edge.attr.comb = "min") #removing duplicates and self-loops
g11 #after simplifying, we obtain a graph with 65 nodes but only 330 edges## IGRAPH 861146a UN-- 65 330 --
## + attr: name (v/c), Alignment (v/c), Gender_Male (v/n), House
## | (v/c), Color (v/c)
## + edges from 861146a (vertex names):
## [1] Regulus Arcturus Black--Sirius Black
## [2] Regulus Arcturus Black--Vincent Crabbe Sr.
## [3] Regulus Arcturus Black--Bellatrix Lestrange
## [4] Regulus Arcturus Black--Lord Voldemort
## [5] Sirius Black --Albus Dumbledore
## [6] Sirius Black --Argus Filch
## [7] Sirius Black --Hermione Granger
## + ... omitted several edgesV(g11) #number of nodes in the network which is equal to 65 - to confirm with g11 analysis## + 65/65 vertices, named, from 861146a:
## [1] Regulus Arcturus Black Sirius Black Lavender Brown
## [4] Cho Chang Vincent Crabbe Sr. Vincent Crabbe
## [7] Bartemius Crouch Sr. Bartemius Crouch Jr. Fleur Delacour
## [10] Cedric Diggory Aberforth Dumbledore Albus Dumbledore
## [13] Dudley Dursley Petunia Dursley Vernon Dursley
## [16] Argus Filch Seamus Finnigan Nicolas Flamel
## [19] Cornelius Fudge Goyle Sr. Gregory Goyle
## [22] Hermione Granger Rubeus Hagrid Igor Karkaroff
## [25] Viktor Krum Bellatrix Lestrange Alice Longbottom
## [28] Frank Longbottom Neville Longbottom Luna Lovegood
## + ... omitted several verticesE(g11) #number of edges in the network which is equal to 330 - ton confirm with g11 analysis## + 330/330 edges from 861146a (vertex names):
## [1] Regulus Arcturus Black--Sirius Black
## [2] Regulus Arcturus Black--Vincent Crabbe Sr.
## [3] Regulus Arcturus Black--Bellatrix Lestrange
## [4] Regulus Arcturus Black--Lord Voldemort
## [5] Sirius Black --Albus Dumbledore
## [6] Sirius Black --Argus Filch
## [7] Sirius Black --Hermione Granger
## [8] Sirius Black --Bellatrix Lestrange
## [9] Sirius Black --Remus Lupin
## [10] Sirius Black --Lucius Malfoy
## + ... omitted several edgesvertex_attr(g11) #describes every column in the graph and the content within that column (all possible values)## $name
## [1] "Regulus Arcturus Black" "Sirius Black"
## [3] "Lavender Brown" "Cho Chang"
## [5] "Vincent Crabbe Sr." "Vincent Crabbe"
## [7] "Bartemius Crouch Sr." "Bartemius Crouch Jr."
## [9] "Fleur Delacour" "Cedric Diggory"
## [11] "Aberforth Dumbledore" "Albus Dumbledore"
## [13] "Dudley Dursley" "Petunia Dursley"
## [15] "Vernon Dursley" "Argus Filch"
## [17] "Seamus Finnigan" "Nicolas Flamel"
## [19] "Cornelius Fudge" "Goyle Sr."
## [21] "Gregory Goyle" "Hermione Granger"
## [23] "Rubeus Hagrid" "Igor Karkaroff"
## [25] "Viktor Krum" "Bellatrix Lestrange"
## [27] "Alice Longbottom" "Frank Longbottom"
## [29] "Neville Longbottom" "Luna Lovegood"
## [31] "Xenophilius Lovegood" "Remus Lupin"
## [33] "Draco Malfoy" "Lucius Malfoy"
## [35] "Narcissa Malfoy" "Olympe Maxime"
## [37] "Minerva McGonagall" "Alastor Moody"
## [39] "Peter Pettigrew" "Harry Potter"
## [41] "James Potter" "Lily Potter"
## [43] "Quirinus Quirrell" "Tom Riddle Sr."
## [45] "Mary Riddle" "Lord Voldemort"
## [47] "Rita Skeeter" "Severus Snape"
## [49] "Nymphadora Tonks" "Dolores Janes Umbridge"
## [51] "Arthur Weasley" "Bill Weasley"
## [53] "Charlie Weasley" "Fred Weasley"
## [55] "George Weasley" "Ginny Weasley"
## [57] "Molly Weasley" "Percy Weasley"
## [59] "Ron Weasley" "Dobby"
## [61] "Fluffy" "Hedwig"
## [63] "Moaning Myrtle" "Aragog"
## [65] "Grawp"
##
## $Alignment
## [1] "Good" "Good" "Good" "Good" "Evil" "Evil" "Good"
## [8] "Evil" "Good" "Good" "Good" "Good" "Neutral" "Neutral"
## [15] "Neutral" "Neutral" "Good" "Good" "Neutral" "Evil" "Evil"
## [22] "Good" "Good" "Neutral" "Good" "Evil" "Good" "Good"
## [29] "Good" "Good" "Good" "Good" "Evil" "Evil" "Evil"
## [36] "Good" "Good" "Good" "Evil" "Good" "Good" "Good"
## [43] "Evil" "Evil" "Good" "Evil" "Good" "Good" "Good"
## [50] "Evil" "Good" "Good" "Good" "Good" "Good" "Good"
## [57] "Good" "Good" "Good" "Good" "Good" "Good" "Good"
## [64] "Neutral" "Good"
##
## $Gender_Male
## [1] 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0
## [36] 0 0 1 1 1 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1
##
## $House
## [1] "Slytherin" "Gryffindor" "Gryffindor" "Ravenclaw" "Slytherin"
## [6] "Slytherin" NA NA "Beauxbatons" "Hufflepuff"
## [11] "Gryffindor" "Gryffindor" "Muggle" "Muggle" "Muggle"
## [16] "Squib" "Gryffindor" "Beauxbatons" NA "Slytherin"
## [21] "Slytherin" "Gryffindor" "Gryffindor" "Durmstrang" "Durmstrang"
## [26] "Slytherin" "Gryffindor" "Gryffindor" "Gryffindor" "Ravenclaw"
## [31] "Ravenclaw" "Gryffindor" "Slytherin" "Slytherin" "Slytherin"
## [36] "Beauxbatons" "Gryffindor" NA "Gryffindor" "Gryffindor"
## [41] "Gryffindor" "Gryffindor" "Ravenclaw" "Muggle" "Muggle"
## [46] "Slytherin" NA "Slytherin" "Hufflepuff" "Slytherin"
## [51] "Gryffindor" "Gryffindor" "Gryffindor" "Gryffindor" "Gryffindor"
## [56] "Gryffindor" "Gryffindor" "Gryffindor" "Gryffindor" NA
## [61] NA NA "Ravenclaw" NA NA
##
## $Color
## [1] "darkgreen" "darkred" "darkred" "blue2"
## [5] "darkgreen" "darkgreen" "white" "white"
## [9] "cadetblue1" "darkgoldenrod1" "darkred" "darkred"
## [13] "bisque" "bisque" "bisque" "cornsilk3"
## [17] "darkred" "cadetblue1" "white" "darkgreen"
## [21] "darkgreen" "darkred" "darkred" "cornsilk4"
## [25] "cornsilk4" "darkgreen" "darkred" "darkred"
## [29] "darkred" "blue2" "blue2" "darkred"
## [33] "darkgreen" "darkgreen" "darkgreen" "cadetblue1"
## [37] "darkred" "white" "darkred" "darkred"
## [41] "darkred" "darkred" "blue2" "bisque"
## [45] "bisque" "darkgreen" "white" "darkgreen"
## [49] "darkgoldenrod1" "darkgreen" "darkred" "darkred"
## [53] "darkred" "darkred" "darkred" "darkred"
## [57] "darkred" "darkred" "darkred" "white"
## [61] "white" "white" "blue2" "white"
## [65] "white"class(g11) #verifying that the graph is now under the class igraph for further analysis## [1] "igraph"Graphing the Network
Graphical representation of the network. The nodes are colored to represent the house to which the node belongs to. For example, characters from Gryffindor are colored in red (see legend for color associations). The Kamada Kawai layout was used becasue it was the best layout when wanting to separate the nodes as much as possible.
As we can see from the graph, most of the characters are Gryffindor, with Slytherin in second place. This isn’t surprising, as the main characters we all know from Harry Potter (Harry Potter, Hermione Granger and Ron Weasley) all belong to the Gryffindors.
In addition, the nodes on the outskirt of the network are the characters with less connections to the other characters. In other words, they know less people than any of the characters in the center of the graph.
color_vertices <- keys %>%
group_by(House, Color) %>%
summarise(n = n()) %>%
filter(!is.na(Color)) #assigning a vertice's color to the its house and color within the dataset
mylayout <- layout.kamada.kawai(g1) #assign chosen layout to use to represent our network
par(mar=c(1,1,3,1)) #modify the layout of the RMarkdown
#plot the initial Harry Potter Network
plot(igraph::simplify(g1,edge.attr.comb = "min"),
layout = mylayout,
legend = TRUE,
vertex.label = V(g1)$Character,
vertex.color = V(g1)$Color,
vertex.frame.color = "gray",
vertex.label.color = "black",
vertex.label.cex = 0.6,
vertex.label.dist=6,
vertex.size = 20,
asp = 0.75,
rescale = FALSE,
ylim = c(-5,3.5),
xlim = c(-5,2))
legend("topleft", legend = c(NA, "", as.character(color_vertices$House)), pch = 19,
col = c(NA, NA, color_vertices$Color), pt.cex = 1.5, cex = 0.65, bty = "n", ncol = 1,
title = "")
legend("topleft", legend = "", cex = 0.95, bty = "n", ncol = 1,
title = "Harry Potter's Network Character Connections")
Alignment Graph of the Characters
This graph gives us an overall view of each character’s alignment. Surprisingly, there is a higher number of “Good” characters amongst the 65 most important. “Evil” and “Neutral” characters seem to have the same proportion.
(The graph is interactive, meaning you can zoom in/out but also can click on specific nodes and move them around. Plus, if the graph does not show in the Markdown, make sure it is in the same zip folder that the R document)
keys_viz<-keys #make a copy of the keys dataframe to avoid writing over the original data
keys_viz$group <- keys$Alignment #group characters based on their alignment
visnet <- visNetwork(keys_viz, edges) #create a visNetwork
#plot the three alignments without any edges because we are interested in the distribution and not the relationship
visnet <- visGroups(visnet, groupname = "Evil", shape = "square",
color = list(background = "tomato", border="black"))
visnet <- visGroups(visnet, groupname = "Good", shape = "dot",
color = list(background = "green", border="black"))
visnet <- visGroups(visnet, groupname = "Neutral", shape = "diamond",
color = list(background = "gray", border="black"))
visLegend(visnet, main="Legend", position="right", ncol=1)Degree Level and Distribution
Degree Level
The actors network has a mean degree value of 10.15. This means that actors within the network are on average directly connected with 10 other actors by virtue of knowing similar people. This seemingly small number is surprising considering this network is comprised of the most important characters in the movies.
It is clear from the histogram charts that the distribution doesn’t follow a normal distribution as it contains a right skew. The distribution in particular follows the so-called “Power Law” in that only a handful of articles have a high degree level.
With regards to standard deviation of the network, it is clear there is a fairly wide distribution (the standard deviation is actually just a bit smaller than the mean). This implies that while the overall average is quite low at approximately 10, there are almost more than a few characters who’s degrees are significantly higher or lower than this figure (shows high dispersity between the network degrees).
To have a better understanding of the first 15 degrees for each graph, we created an additional histogram representing the splitting of the degree distribution between 0 and 15 degrees for characters. From this, we can see that from 0 to 15 degrees, the frequencies are better distributed. Only degree 4 is considerably more frequent than the others.
deg<-igraph::degree(g11, mode = "total")
hist(deg, main="Histogram of Node Degree For Characters", ylim=c(0,30), col = "tomato") #plotting histogram of degree level frequencies
summary(deg) #gain information on the mean## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.00 4.00 7.00 10.15 13.00 48.00sd(deg) #gain information on the standard deviation## [1] 9.569206#when looking at the first 3 bars more in depth; range of 0 to 15 degrees
hist_1<-hist(deg, main="Histogram of Node Degree For Characters From 0-15 Degrees", xlim=c(0,15), ylim=c(0,30), breaks=888, col = "tomato") 
Degree Distribution
The degree distribution looks like a continuous logarithmic shaped where the cumulative frequency increases at a fast pace and then stagnates; it is discontinued at the beginning and then becomes constant when it becomes closer to 1. This confirms that most characters are within degree levels of 22. We can infer, that these characters, may represent secondary or tertiary roles in the movies, with only a few main character to the story line.
deg.dist <- igraph::degree_distribution(g11, cumulative=T, mode="all")
plot( x=0:max(deg), y=1-deg.dist, pch=19, cex=1.2, col="orange", xlab="Degree", ylab="Cumulative Frequency for Characters")
summary(deg.dist)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.01538 0.06154 0.07692 0.22763 0.32308 1.00000Network Based on Level of Degree
Graphical representation of the network where the nodes are still colored to represent the house to which the node belongs to. However, now, the node sizes are based on the level of degree each character has (higher levels of degree have bigger node sizes).
As we can see from the graph, not surprisingly, the characters with the highest levels of degree are the main protagonists from the story line. This not only includes Harry, Hermione and Ron, but also Dumbledore and Voldemort. Later in the analysis, we will see which character has the highest degree between them.
par(mar=c(1,1,3,1)) #layout of the RMarkdown
plot(igraph::simplify(g1,edge.attr.comb = "min"),
layout = mylayout,
legend = TRUE,
vertex.label = V(g1)$Character,
vertex.color = V(g1)$Color,
vertex.frame.color = "gray",
vertex.label.color = "black",
vertex.size = deg*1.3,
vertex.label.cex = 0.6,
vertex.label.dist=3,
asp = 0.75,
rescale = FALSE,
ylim = c(-5,3),
xlim = c(-5,2))
legend("topleft", legend = c(NA, "", as.character(color_vertices$House)), pch = 19,
col = c(NA, NA, color_vertices$Color), pt.cex = 1.5, cex = 0.65, bty = "n", ncol = 1,
title = "")
legend("topleft", legend = "", cex = 0.95, bty = "n", ncol = 1,
title = "Harry Potter's Character Level of Degree")
Network Properties
Connected Component (Graph Theory)
In graph theory, a connected component (or just component) of an undirected graph is a graph in which any two vertices are connected to each other by paths. The cohesion of a graph is the vertex connectivity of the graph. This is calculated by cohesion. In our case, we can see that our graph, from prior visualization was connected, but this result is also supported by the cohesion equation, which equals to 1. Thus, our whole network is confirmed as connected and every character in our network is somewhat connected to each other.
cohesion(g11, checks = TRUE) #calculate if our whole network is connected or not## [1] 1Network Average Path Length
The average path length describes the average distance between any two vertices (nodes).
The average path length of the Harry Potter network has a value of 2.02. This suggests that the network is on average not very dispersed, meaning that any two randomly selected characters of Harry Potter are closely related. This would make sense as this data represents the most important characters within the movies.
mean_distance(g11, directed=F) #calculate the average path length in an undirected graph## [1] 2.028365Graph Theory: Dijkstra
Now that we have calculated the average path length, the Dijkstra algorithm identifies the shortest paths in the network. Here are the steps to follow in order to obtain these paths:
First, to run the algorithm, we had to compute the Adjacency Matrix.
g11_edgelist<-as_edgelist(g11) #the adjacency matrix can only be calculated from a list of edges (conversts the grah's edges to a list)
g11_ajd<-get.adjacency(graph.edgelist(as.matrix(g11_edgelist), directed=FALSE)) #compute adjacency matrix
g11_ajd #by calling the adjacency matrix, we can see which characters have a connection (1) and when they don't (0) - sample of the matrix## 65 x 65 sparse Matrix of class "dgCMatrix"## [[ suppressing 65 column names 'Regulus Arcturus Black', 'Sirius Black', 'Vincent Crabbe Sr.' ... ]]##
## Regulus Arcturus Black . 1 1 1 1 . . . . . . . . . . . . . . . . . . . . .
## Sirius Black 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .
## Vincent Crabbe Sr. 1 . . 1 1 . . . . 1 . . . 1 . . . 1 . . . . . . . .
## Bellatrix Lestrange 1 1 1 . 1 1 . . . 1 1 . . . 1 . . . 1 . . . . 1 1 .
## Lord Voldemort 1 1 1 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 . 1 1 1 1 1 .
## Albus Dumbledore . 1 . 1 1 . 1 1 1 . . 1 1 1 1 1 1 1 . 1 1 1 1 1 1 .
## Argus Filch . 1 . . . 1 . 1 . . . 1 . 1 1 1 . 1 . . 1 1 . . 1 .
## Hermione Granger . 1 . . 1 1 1 . . . . 1 1 1 1 . . 1 . 1 1 1 1 1 1 1
## Remus Lupin . 1 . . 1 1 . . . . . 1 . 1 1 1 . . 1 . 1 1 1 . 1 .
## Lucius Malfoy . 1 1 1 1 . . . . . 1 . . . 1 . . . . 1 . . . 1 1 .
## Narcissa Malfoy . 1 . 1 1 . . . . 1 . . . . . . . . . . . . . . . .
## Minerva McGonagall . 1 . . 1 1 1 1 1 . . . . . 1 . . . . . . . . . 1 .
## Alastor Moody . 1 . . 1 1 . 1 . . . . . . 1 . . . . 1 . . . 1 1 .
## Peter Pettigrew . 1 1 . 1 1 1 1 1 . . . . . 1 1 1 . . . 1 1 1 . 1 .
## Harry Potter . 1 . 1 1 1 1 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 1 1 1 .
## James Potter . 1 . . 1 1 1 . 1 . . . . 1 1 . 1 . . . . . . . . .
## Lily Potter . 1 . . 1 1 . . . . . . . 1 1 1 . 1 . . . . . . . .
## Severus Snape . 1 1 . 1 1 1 1 . . . . . . 1 . 1 . . . 1 1 1 . 1 .
## Nymphadora Tonks . 1 . 1 1 . . . 1 . . . . . . . . . . . . . . . . .
## Arthur Weasley . 1 . . . 1 . 1 . 1 . . 1 . 1 . . . . . 1 1 1 1 1 .
## Fred Weasley . 1 . . 1 1 1 1 1 . . . . 1 1 . . 1 . 1 . 1 1 1 1 .
## George Weasley . 1 . . 1 1 1 1 1 . . . . 1 1 . . 1 . 1 1 . 1 1 1 .
## Ginny Weasley . 1 . . 1 1 . 1 1 . . . . 1 1 . . 1 . 1 1 1 . 1 1 .
## Molly Weasley . 1 . 1 1 1 . 1 . 1 . . 1 . 1 . . . . 1 1 1 1 . 1 .
## Ron Weasley . 1 . 1 1 1 1 1 1 1 . 1 1 1 1 . . 1 . 1 1 1 1 1 . 1
## Lavender Brown . . . . . . . 1 . . . . . . . . . . . . . . . . 1 .
## Seamus Finnigan . . . . . . . 1 . . . . . . 1 . . . . . . . 1 . 1 1
## Neville Longbottom . . . . 1 . . 1 . . . . . . 1 . . . . . . . . . 1 1
## Cho Chang . . . . 1 . . 1 . . . . . . 1 . . . . . . . . . 1 .
## Cedric Diggory . . . . 1 . . . . . . . . . 1 . . . . . . . . . . .
## Vincent Crabbe . . 1 . . . . 1 . . . . . . 1 . . . . . . . . . 1 .
## Bartemius Crouch Sr. . . 1 . . . . . . . . . . . . . . . . . . . . . . .
## Goyle Sr. . . 1 . 1 . . . . . . . . . . . . . . . . . . . . .
## Gregory Goyle . . . . 1 . . 1 . . . . . . 1 . . . . . . . . . 1 .
## Draco Malfoy . . . . . . . 1 . 1 1 . . . 1 . . . . . . . . . 1 .
## Dolores Janes Umbridge . . . . . 1 1 1 . . . 1 . . 1 . . . . . . . . . 1 .
## Bartemius Crouch Jr. . . . . 1 . . 1 . . . 1 1 1 1 . . 1 . . . . . . . .
## Igor Karkaroff . . . . 1 . . . . . . . . . . . . . . . . . . . . .
## Percy Weasley . . . . . 1 . . . . . . . . . . . . . 1 1 1 1 1 1 .
## Fleur Delacour . . . 1 . . . 1 . . . . 1 . 1 . . . . . . . 1 1 1 .
## Viktor Krum . . . . . . . 1 . . . . . . 1 . . . . . . . . . . .
## Olympe Maxime . . . . . 1 . . . . . . . . . . . . . . . . . . . .
## Bill Weasley . . . . 1 1 . 1 . . . . . . 1 . . . . 1 1 1 1 1 1 .
## Aberforth Dumbledore . . . . . 1 . 1 . . . . . . 1 . . . . . . . . . 1 .
## Dobby . . . 1 . . . 1 . 1 . . . . 1 . . . . . . . . . 1 .
## Petunia Dursley . . . . . 1 . . . . . . . . 1 . 1 . . . . . . . . .
## Vernon Dursley . . . . . 1 . . . . . . . . 1 . . . . . . . . . . .
## Nicolas Flamel . . . . . 1 . . . . . . . . . . . . . . . . . . . .
## Cornelius Fudge . . . . . 1 . . . 1 . . . . 1 . . . . 1 . . . . . .
## Rubeus Hagrid . . . . 1 1 1 1 . . . 1 . . 1 . . 1 . . . . . . 1 .
## Quirinus Quirrell . . . . 1 1 . 1 . . . . . . 1 . . . . . . . . . . .
## Rita Skeeter . . . . . 1 . . . . . . . . 1 . . . . . . . . . . .
## Charlie Weasley . . . . 1 1 . 1 . . . . . . 1 . . . . 1 1 1 1 1 1 .
## Dudley Dursley . . . . . . . . . . . . . . 1 . . . . . . . . . . .
## Luna Lovegood . . . . . . . 1 . . . . . . 1 . . . . . . . . . 1 .
## Fluffy . . . . . . . 1 . . . . . . 1 . . . . . . . . . 1 .
## Aragog . . . . . . . 1 . . . . . . 1 . . . . . . . . . 1 .
## Grawp . . . . . . . 1 . . . . . . 1 . . . . . . . . . 1 .
## Alice Longbottom . . . . . . . . . . . . . . . . . . . . . . . . . .
## Frank Longbottom . . . . 1 . . . . . . . . . . . . . . . . . . . . .
## Xenophilius Lovegood . . . . . . . . . . . . . . . . . . . . . . . . . .
## Hedwig . . . . . . . . . . . . . . 1 . . . . . . . . . . .
## Moaning Myrtle . . . . . . . . . . . . . . 1 . . . . . . . . . . .
## Tom Riddle Sr. . . . . 1 . . . . . . . . . . . . . . . . . . . . .
## Mary Riddle . . . . 1 . . . . . . . . . . . . . . . . . . . . .
##
## Regulus Arcturus Black . . . . . . . . . . . . . . . . . . . . . . . . . .
## Sirius Black . . . . . . . . . . . . . . . . . . . . . . . . . .
## Vincent Crabbe Sr. . . . . 1 1 1 . . . . . . . . . . . . . . . . . . .
## Bellatrix Lestrange . . . . . . . . . . . . . 1 . . . . 1 . . . . . . .
## Lord Voldemort . 1 1 1 . . 1 1 . . 1 1 . . . . 1 . . . . . . 1 1 .
## Albus Dumbledore . . . . . . . . . 1 . . 1 . . 1 1 1 . 1 1 1 1 1 1 1
## Argus Filch . . . . . . . . . 1 . . . . . . . . . . . . . 1 . .
## Hermione Granger 1 1 1 . 1 . . 1 1 1 1 . . 1 1 . 1 1 1 . . . . 1 1 .
## Remus Lupin . . . . . . . . . . . . . . . . . . . . . . . . . .
## Lucius Malfoy . . . . . . . . 1 . . . . . . . . . 1 . . . 1 . . .
## Narcissa Malfoy . . . . . . . . 1 . . . . . . . . . . . . . . . . .
## Minerva McGonagall . . . . . . . . . 1 1 . . . . . . . . . . . . 1 . .
## Alastor Moody . . . . . . . . . . 1 . . 1 . . . . . . . . . . . .
## Peter Pettigrew . . . . . . . . . . 1 . . . . . . . . . . . . . . .
## Harry Potter 1 1 1 1 1 . . 1 1 1 1 . . 1 1 . 1 1 1 1 1 . 1 1 1 1
## James Potter . . . . . . . . . . . . . . . . . . . . . . . . . .
## Lily Potter . . . . . . . . . . . . . . . . . . . 1 . . . . . .
## Severus Snape . . . . . . . . . . 1 . . . . . . . . . . . . 1 . .
## Nymphadora Tonks . . . . . . . . . . . . . . . . . . . . . . . . . .
## Arthur Weasley . . . . . . . . . . . . 1 . . . 1 . . . . . 1 . . .
## Fred Weasley . . . . . . . . . . . . 1 . . . 1 . . . . . . . . .
## George Weasley . . . . . . . . . . . . 1 . . . 1 . . . . . . . . .
## Ginny Weasley 1 . . . . . . . . . . . 1 1 . . 1 . . . . . . . . .
## Molly Weasley . . . . . . . . . . . . 1 1 . . 1 . . . . . . . . .
## Ron Weasley 1 1 1 . 1 . . 1 1 1 . . 1 1 . . 1 1 1 . . . . 1 . .
## Lavender Brown 1 1 . . . . . . . . . . . . . . . . . . . . . . . .
## Seamus Finnigan . 1 1 . . . . . . . . . . . . . . . . . . . . . . .
## Neville Longbottom 1 . 1 . . . . . 1 . 1 . . 1 . . . . . . . . . 1 . .
## Cho Chang 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . .
## Cedric Diggory . . 1 . . . . . . . . . . . 1 . . . . . . . . . . .
## Vincent Crabbe . . . . . . . 1 1 1 . . . . . . . . . . . . . . . .
## Bartemius Crouch Sr. . . . . . . . . . . 1 1 1 . . . . . . . . . . . . .
## Goyle Sr. . . . . . . . 1 . . . . . . . . . . . . . . . . . .
## Gregory Goyle . . . . 1 . 1 . 1 . . . . . . . . . . . . . . . . .
## Draco Malfoy . 1 . . 1 . . 1 . . . . . . . . . . 1 . . . . . . .
## Dolores Janes Umbridge . . . . 1 . . . . . . . . . . . . . . . . . 1 . . .
## Bartemius Crouch Jr. . 1 . . . 1 . . . . . . . . . . . . . . . . . . . .
## Igor Karkaroff . . . . . 1 . . . . . . . . 1 1 . . . . . . . . . .
## Percy Weasley . . . . . 1 . . . . . . . . . . 1 . . . . . 1 . . .
## Fleur Delacour . 1 . . . . . . . . . . . . 1 1 1 . . . . . . . . .
## Viktor Krum . . . 1 . . . . . . . 1 . 1 . . . . . . . . . . . .
## Olympe Maxime . . . . . . . . . . . 1 . 1 . . . . . . . . . 1 . .
## Bill Weasley . . . . . . . . . . . . 1 1 . . . . . . . . . . . .
## Aberforth Dumbledore . . . . . . . . . . . . . . . . . . 1 . . . . . . .
## Dobby . . . . . . . . 1 . . . . . . . . 1 . . . . . . . .
## Petunia Dursley . . . . . . . . . . . . . . . . . . . . 1 . . . . .
## Vernon Dursley . . . . . . . . . . . . . . . . . . . 1 . . . . . .
## Nicolas Flamel . . . . . . . . . . . . . . . . . . . . . . . . . .
## Cornelius Fudge . . . . . . . . . 1 . . 1 . . . . . . . . . . . . .
## Rubeus Hagrid . 1 . . . . . . . . . . . . . 1 . . . . . . . . . .
## Quirinus Quirrell . . . . . . . . . . . . . . . . . . . . . . . . . .
## Rita Skeeter . . . . . . . . . . . . . . . . . . . . . . . . . .
## Charlie Weasley . . . . . . . . . . . . 1 . . . 1 . . . . . . . . .
## Dudley Dursley . . . . . . . . . . . . . . . . . . . 1 1 . . 1 . .
## Luna Lovegood 1 1 . . . . . . . . . . . . . . . . . . . . . . . .
## Fluffy . . . . . . . . . . . . . . . . . . . . . . . 1 . .
## Aragog . . . . . . . . . . . . . . . . . . . . . . . 1 . .
## Grawp . . . . . . . . . . . . . . . . . . . . . . . 1 . .
## Alice Longbottom . 1 . . . . . . . . . . . . . . . . . . . . . . . .
## Frank Longbottom . 1 . . . . . . . . . . . . . . . . . . . . . . . .
## Xenophilius Lovegood . . . . . . . . . . . . . . . . . . . . . . . . . .
## Hedwig . . . . . . . . . . . . . . . . . . . . . . . . . .
## Moaning Myrtle . . . . . . . . . . . . . . . . . . . . . . . . . .
## Tom Riddle Sr. . . . . . . . . . . . . . . . . . . . . . . . . . .
## Mary Riddle . . . . . . . . . . . . . . . . . . . . . . . . . .
##
## Regulus Arcturus Black . . . . . . . . . . . . .
## Sirius Black . . . . . . . . . . . . .
## Vincent Crabbe Sr. . . . . . . . . . . . . .
## Bellatrix Lestrange . . . . . . . . . . . . .
## Lord Voldemort 1 . . . . . . 1 . . . 1 1
## Albus Dumbledore 1 . . . . . . . . . . . .
## Argus Filch . . . . . . . . . . . . .
## Hermione Granger 1 . 1 1 1 1 . . . . . . .
## Remus Lupin . . . . . . . . . . . . .
## Lucius Malfoy . . . . . . . . . . . . .
## Narcissa Malfoy . . . . . . . . . . . . .
## Minerva McGonagall . . . . . . . . . . . . .
## Alastor Moody . . . . . . . . . . . . .
## Peter Pettigrew . . . . . . . . . . . . .
## Harry Potter 1 1 1 1 1 1 . . . 1 1 . .
## James Potter . . . . . . . . . . . . .
## Lily Potter . . . . . . . . . . . . .
## Severus Snape . . . . . . . . . . . . .
## Nymphadora Tonks . . . . . . . . . . . . .
## Arthur Weasley 1 . . . . . . . . . . . .
## Fred Weasley 1 . . . . . . . . . . . .
## George Weasley 1 . . . . . . . . . . . .
## Ginny Weasley 1 . . . . . . . . . . . .
## Molly Weasley 1 . . . . . . . . . . . .
## Ron Weasley 1 . 1 1 1 1 . . . . . . .
## Lavender Brown . . . . . . . . . . . . .
## Seamus Finnigan . . 1 . . . . . . . . . .
## Neville Longbottom . . 1 . . . 1 1 . . . . .
## Cho Chang . . . . . . . . . . . . .
## Cedric Diggory . . . . . . . . . . . . .
## Vincent Crabbe . . . . . . . . . . . . .
## Bartemius Crouch Sr. . . . . . . . . . . . . .
## Goyle Sr. . . . . . . . . . . . . .
## Gregory Goyle . . . . . . . . . . . . .
## Draco Malfoy . . . . . . . . . . . . .
## Dolores Janes Umbridge . . . . . . . . . . . . .
## Bartemius Crouch Jr. . . . . . . . . . . . . .
## Igor Karkaroff . . . . . . . . . . . . .
## Percy Weasley 1 . . . . . . . . . . . .
## Fleur Delacour . . . . . . . . . . . . .
## Viktor Krum . . . . . . . . . . . . .
## Olympe Maxime . . . . . . . . . . . . .
## Bill Weasley 1 . . . . . . . . . . . .
## Aberforth Dumbledore . . . . . . . . . . . . .
## Dobby . . . . . . . . . . . . .
## Petunia Dursley . 1 . . . . . . . . . . .
## Vernon Dursley . 1 . . . . . . . . . . .
## Nicolas Flamel . . . . . . . . . . . . .
## Cornelius Fudge . . . . . . . . . . . . .
## Rubeus Hagrid . 1 . 1 1 1 . . . . . . .
## Quirinus Quirrell . . . . . . . . . . . . .
## Rita Skeeter . . . . . . . . . . . . .
## Charlie Weasley . . . . . . . . . . . . .
## Dudley Dursley . . . . . . . . . . . . .
## Luna Lovegood . . . . . . . . 1 . . . .
## Fluffy . . . . . . . . . . . . .
## Aragog . . . . . . . . . . . . .
## Grawp . . . . . . . . . . . . .
## Alice Longbottom . . . . . . . 1 . . . . .
## Frank Longbottom . . . . . . 1 . . . . . .
## Xenophilius Lovegood . . 1 . . . . . . . . . .
## Hedwig . . . . . . . . . . . . .
## Moaning Myrtle . . . . . . . . . . . . .
## Tom Riddle Sr. . . . . . . . . . . . . 1
## Mary Riddle . . . . . . . . . . . 1 .Then, we were able to run Dijkstra, from which we obtained the matrix of the shortest paths. On average, the shortest paths are of 2, which supports the fact that our average path length is equal to 2. We can see that there is also only one shortest path equivalent to 4, which we can assume is our diameter (this will be confirmed below).
graph_ajd<-graph.adjacency(g11_ajd) #convert matrix adjacency to a graph for which we can then obtain the shortest paths matrix (below) when using the dijkstra algorithm
s.paths <- shortest.paths(graph_ajd, algorithm = "dijkstra")
s.paths## Regulus Arcturus Black Sirius Black
## Regulus Arcturus Black 0 1
## Sirius Black 1 0
## Vincent Crabbe Sr. 1 2
## Bellatrix Lestrange 1 1
## Lord Voldemort 1 1
## Albus Dumbledore 2 1
## Argus Filch 2 1
## Hermione Granger 2 1
## Remus Lupin 2 1
## Lucius Malfoy 2 1
## Narcissa Malfoy 2 1
## Minerva McGonagall 2 1
## Alastor Moody 2 1
## Peter Pettigrew 2 1
## Harry Potter 2 1
## James Potter 2 1
## Lily Potter 2 1
## Severus Snape 2 1
## Nymphadora Tonks 2 1
## Arthur Weasley 2 1
## Fred Weasley 2 1
## George Weasley 2 1
## Ginny Weasley 2 1
## Molly Weasley 2 1
## Ron Weasley 2 1
## Lavender Brown 3 2
## Seamus Finnigan 3 2
## Neville Longbottom 2 2
## Cho Chang 2 2
## Cedric Diggory 2 2
## Vincent Crabbe 2 2
## Bartemius Crouch Sr. 2 3
## Goyle Sr. 2 2
## Gregory Goyle 2 2
## Draco Malfoy 3 2
## Dolores Janes Umbridge 3 2
## Bartemius Crouch Jr. 2 2
## Igor Karkaroff 2 2
## Percy Weasley 3 2
## Fleur Delacour 2 2
## Viktor Krum 3 2
## Olympe Maxime 3 2
## Bill Weasley 2 2
## Aberforth Dumbledore 3 2
## Dobby 2 2
## Petunia Dursley 3 2
## Vernon Dursley 3 2
## Nicolas Flamel 3 2
## Cornelius Fudge 3 2
## Rubeus Hagrid 2 2
## Quirinus Quirrell 2 2
## Rita Skeeter 3 2
## Charlie Weasley 2 2
## Dudley Dursley 3 2
## Luna Lovegood 3 2
## Fluffy 3 2
## Aragog 3 2
## Grawp 3 2
## Alice Longbottom 3 3
## Frank Longbottom 2 2
## Xenophilius Lovegood 4 3
## Hedwig 3 2
## Moaning Myrtle 3 2
## Tom Riddle Sr. 2 2
## Mary Riddle 2 2
## Vincent Crabbe Sr. Bellatrix Lestrange
## Regulus Arcturus Black 1 1
## Sirius Black 2 1
## Vincent Crabbe Sr. 0 1
## Bellatrix Lestrange 1 0
## Lord Voldemort 1 1
## Albus Dumbledore 2 1
## Argus Filch 2 2
## Hermione Granger 2 2
## Remus Lupin 2 2
## Lucius Malfoy 1 1
## Narcissa Malfoy 2 1
## Minerva McGonagall 2 2
## Alastor Moody 2 2
## Peter Pettigrew 1 2
## Harry Potter 2 1
## James Potter 2 2
## Lily Potter 2 2
## Severus Snape 1 2
## Nymphadora Tonks 2 1
## Arthur Weasley 2 2
## Fred Weasley 2 2
## George Weasley 2 2
## Ginny Weasley 2 2
## Molly Weasley 2 1
## Ron Weasley 2 1
## Lavender Brown 3 2
## Seamus Finnigan 3 2
## Neville Longbottom 2 2
## Cho Chang 2 2
## Cedric Diggory 2 2
## Vincent Crabbe 1 2
## Bartemius Crouch Sr. 1 2
## Goyle Sr. 1 2
## Gregory Goyle 2 2
## Draco Malfoy 2 2
## Dolores Janes Umbridge 2 2
## Bartemius Crouch Jr. 2 2
## Igor Karkaroff 2 2
## Percy Weasley 2 2
## Fleur Delacour 2 1
## Viktor Krum 3 2
## Olympe Maxime 3 2
## Bill Weasley 2 2
## Aberforth Dumbledore 3 2
## Dobby 2 1
## Petunia Dursley 3 2
## Vernon Dursley 3 2
## Nicolas Flamel 3 2
## Cornelius Fudge 2 2
## Rubeus Hagrid 2 2
## Quirinus Quirrell 2 2
## Rita Skeeter 3 2
## Charlie Weasley 2 2
## Dudley Dursley 3 2
## Luna Lovegood 3 2
## Fluffy 3 2
## Aragog 3 2
## Grawp 3 2
## Alice Longbottom 3 3
## Frank Longbottom 2 2
## Xenophilius Lovegood 4 3
## Hedwig 3 2
## Moaning Myrtle 3 2
## Tom Riddle Sr. 2 2
## Mary Riddle 2 2
## Lord Voldemort Albus Dumbledore Argus Filch
## Regulus Arcturus Black 1 2 2
## Sirius Black 1 1 1
## Vincent Crabbe Sr. 1 2 2
## Bellatrix Lestrange 1 1 2
## Lord Voldemort 0 1 2
## Albus Dumbledore 1 0 1
## Argus Filch 2 1 0
## Hermione Granger 1 1 1
## Remus Lupin 1 1 2
## Lucius Malfoy 1 2 2
## Narcissa Malfoy 1 2 2
## Minerva McGonagall 1 1 1
## Alastor Moody 1 1 2
## Peter Pettigrew 1 1 1
## Harry Potter 1 1 1
## James Potter 1 1 1
## Lily Potter 1 1 2
## Severus Snape 1 1 1
## Nymphadora Tonks 1 2 2
## Arthur Weasley 2 1 2
## Fred Weasley 1 1 1
## George Weasley 1 1 1
## Ginny Weasley 1 1 2
## Molly Weasley 1 1 2
## Ron Weasley 1 1 1
## Lavender Brown 2 2 2
## Seamus Finnigan 2 2 2
## Neville Longbottom 1 2 2
## Cho Chang 1 2 2
## Cedric Diggory 1 2 2
## Vincent Crabbe 2 2 2
## Bartemius Crouch Sr. 2 2 3
## Goyle Sr. 1 2 3
## Gregory Goyle 1 2 2
## Draco Malfoy 2 2 2
## Dolores Janes Umbridge 2 1 1
## Bartemius Crouch Jr. 1 2 2
## Igor Karkaroff 1 2 3
## Percy Weasley 2 1 2
## Fleur Delacour 2 2 2
## Viktor Krum 2 2 2
## Olympe Maxime 2 1 2
## Bill Weasley 1 1 2
## Aberforth Dumbledore 2 1 2
## Dobby 2 2 2
## Petunia Dursley 2 1 2
## Vernon Dursley 2 1 2
## Nicolas Flamel 2 1 2
## Cornelius Fudge 2 1 2
## Rubeus Hagrid 1 1 1
## Quirinus Quirrell 1 1 2
## Rita Skeeter 2 1 2
## Charlie Weasley 1 1 2
## Dudley Dursley 2 2 2
## Luna Lovegood 2 2 2
## Fluffy 2 2 2
## Aragog 2 2 2
## Grawp 2 2 2
## Alice Longbottom 2 3 3
## Frank Longbottom 1 2 3
## Xenophilius Lovegood 3 3 3
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 1 2 3
## Mary Riddle 1 2 3
## Hermione Granger Remus Lupin Lucius Malfoy
## Regulus Arcturus Black 2 2 2
## Sirius Black 1 1 1
## Vincent Crabbe Sr. 2 2 1
## Bellatrix Lestrange 2 2 1
## Lord Voldemort 1 1 1
## Albus Dumbledore 1 1 2
## Argus Filch 1 2 2
## Hermione Granger 0 2 2
## Remus Lupin 2 0 2
## Lucius Malfoy 2 2 0
## Narcissa Malfoy 2 2 1
## Minerva McGonagall 1 1 2
## Alastor Moody 1 2 2
## Peter Pettigrew 1 1 2
## Harry Potter 1 1 1
## James Potter 2 1 2
## Lily Potter 2 2 2
## Severus Snape 1 2 2
## Nymphadora Tonks 2 1 2
## Arthur Weasley 1 2 1
## Fred Weasley 1 1 2
## George Weasley 1 1 2
## Ginny Weasley 1 1 2
## Molly Weasley 1 2 1
## Ron Weasley 1 1 1
## Lavender Brown 1 2 2
## Seamus Finnigan 1 2 2
## Neville Longbottom 1 2 2
## Cho Chang 1 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 1 2 2
## Bartemius Crouch Sr. 2 3 2
## Goyle Sr. 2 2 2
## Gregory Goyle 1 2 2
## Draco Malfoy 1 2 1
## Dolores Janes Umbridge 1 2 2
## Bartemius Crouch Jr. 1 2 2
## Igor Karkaroff 2 2 2
## Percy Weasley 2 2 2
## Fleur Delacour 1 2 2
## Viktor Krum 1 2 2
## Olympe Maxime 2 2 3
## Bill Weasley 1 2 2
## Aberforth Dumbledore 1 2 2
## Dobby 1 2 1
## Petunia Dursley 2 2 2
## Vernon Dursley 2 2 2
## Nicolas Flamel 2 2 3
## Cornelius Fudge 2 2 1
## Rubeus Hagrid 1 2 2
## Quirinus Quirrell 1 2 2
## Rita Skeeter 2 2 2
## Charlie Weasley 1 2 2
## Dudley Dursley 2 2 2
## Luna Lovegood 1 2 2
## Fluffy 1 2 2
## Aragog 1 2 2
## Grawp 1 2 2
## Alice Longbottom 2 3 3
## Frank Longbottom 2 2 2
## Xenophilius Lovegood 2 3 3
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 2 2 2
## Mary Riddle 2 2 2
## Narcissa Malfoy Minerva McGonagall Alastor Moody
## Regulus Arcturus Black 2 2 2
## Sirius Black 1 1 1
## Vincent Crabbe Sr. 2 2 2
## Bellatrix Lestrange 1 2 2
## Lord Voldemort 1 1 1
## Albus Dumbledore 2 1 1
## Argus Filch 2 1 2
## Hermione Granger 2 1 1
## Remus Lupin 2 1 2
## Lucius Malfoy 1 2 2
## Narcissa Malfoy 0 2 2
## Minerva McGonagall 2 0 2
## Alastor Moody 2 2 0
## Peter Pettigrew 2 2 2
## Harry Potter 2 1 1
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 2 2 2
## Arthur Weasley 2 2 1
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 2 2 2
## Molly Weasley 2 2 1
## Ron Weasley 2 1 1
## Lavender Brown 3 2 2
## Seamus Finnigan 3 2 2
## Neville Longbottom 2 2 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 2 2
## Bartemius Crouch Sr. 3 2 2
## Goyle Sr. 2 2 2
## Gregory Goyle 2 2 2
## Draco Malfoy 1 2 2
## Dolores Janes Umbridge 3 1 2
## Bartemius Crouch Jr. 2 1 1
## Igor Karkaroff 2 2 2
## Percy Weasley 3 2 2
## Fleur Delacour 2 2 1
## Viktor Krum 3 2 2
## Olympe Maxime 3 2 2
## Bill Weasley 2 2 2
## Aberforth Dumbledore 3 2 2
## Dobby 2 2 2
## Petunia Dursley 3 2 2
## Vernon Dursley 3 2 2
## Nicolas Flamel 3 2 2
## Cornelius Fudge 2 2 2
## Rubeus Hagrid 2 1 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 3 2 2
## Charlie Weasley 2 2 2
## Dudley Dursley 3 2 2
## Luna Lovegood 3 2 2
## Fluffy 3 2 2
## Aragog 3 2 2
## Grawp 3 2 2
## Alice Longbottom 3 3 3
## Frank Longbottom 2 2 2
## Xenophilius Lovegood 4 3 3
## Hedwig 3 2 2
## Moaning Myrtle 3 2 2
## Tom Riddle Sr. 2 2 2
## Mary Riddle 2 2 2
## Peter Pettigrew Harry Potter James Potter
## Regulus Arcturus Black 2 2 2
## Sirius Black 1 1 1
## Vincent Crabbe Sr. 1 2 2
## Bellatrix Lestrange 2 1 2
## Lord Voldemort 1 1 1
## Albus Dumbledore 1 1 1
## Argus Filch 1 1 1
## Hermione Granger 1 1 2
## Remus Lupin 1 1 1
## Lucius Malfoy 2 1 2
## Narcissa Malfoy 2 2 2
## Minerva McGonagall 2 1 2
## Alastor Moody 2 1 2
## Peter Pettigrew 0 1 1
## Harry Potter 1 0 1
## James Potter 1 1 0
## Lily Potter 1 1 1
## Severus Snape 2 1 2
## Nymphadora Tonks 2 2 2
## Arthur Weasley 2 1 2
## Fred Weasley 1 1 2
## George Weasley 1 1 2
## Ginny Weasley 1 1 2
## Molly Weasley 2 1 2
## Ron Weasley 1 1 2
## Lavender Brown 2 2 3
## Seamus Finnigan 2 1 2
## Neville Longbottom 2 1 2
## Cho Chang 2 1 2
## Cedric Diggory 2 1 2
## Vincent Crabbe 2 1 2
## Bartemius Crouch Sr. 2 2 3
## Goyle Sr. 2 2 2
## Gregory Goyle 2 1 2
## Draco Malfoy 2 1 2
## Dolores Janes Umbridge 2 1 2
## Bartemius Crouch Jr. 1 1 2
## Igor Karkaroff 2 2 2
## Percy Weasley 2 2 2
## Fleur Delacour 2 1 2
## Viktor Krum 2 1 2
## Olympe Maxime 2 2 2
## Bill Weasley 2 1 2
## Aberforth Dumbledore 2 1 2
## Dobby 2 1 2
## Petunia Dursley 2 1 2
## Vernon Dursley 2 1 2
## Nicolas Flamel 2 2 2
## Cornelius Fudge 2 1 2
## Rubeus Hagrid 2 1 2
## Quirinus Quirrell 2 1 2
## Rita Skeeter 2 1 2
## Charlie Weasley 2 1 2
## Dudley Dursley 2 1 2
## Luna Lovegood 2 1 2
## Fluffy 2 1 2
## Aragog 2 1 2
## Grawp 2 1 2
## Alice Longbottom 3 2 3
## Frank Longbottom 2 2 2
## Xenophilius Lovegood 3 2 3
## Hedwig 2 1 2
## Moaning Myrtle 2 1 2
## Tom Riddle Sr. 2 2 2
## Mary Riddle 2 2 2
## Lily Potter Severus Snape Nymphadora Tonks
## Regulus Arcturus Black 2 2 2
## Sirius Black 1 1 1
## Vincent Crabbe Sr. 2 1 2
## Bellatrix Lestrange 2 2 1
## Lord Voldemort 1 1 1
## Albus Dumbledore 1 1 2
## Argus Filch 2 1 2
## Hermione Granger 2 1 2
## Remus Lupin 2 2 1
## Lucius Malfoy 2 2 2
## Narcissa Malfoy 2 2 2
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 1 2 2
## Harry Potter 1 1 2
## James Potter 1 2 2
## Lily Potter 0 1 2
## Severus Snape 1 0 2
## Nymphadora Tonks 2 2 0
## Arthur Weasley 2 2 2
## Fred Weasley 2 1 2
## George Weasley 2 1 2
## Ginny Weasley 2 1 2
## Molly Weasley 2 2 2
## Ron Weasley 2 1 2
## Lavender Brown 3 2 3
## Seamus Finnigan 2 2 3
## Neville Longbottom 2 2 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 2 3
## Bartemius Crouch Sr. 3 2 3
## Goyle Sr. 2 2 2
## Gregory Goyle 2 2 2
## Draco Malfoy 2 2 3
## Dolores Janes Umbridge 2 2 3
## Bartemius Crouch Jr. 2 1 2
## Igor Karkaroff 2 2 2
## Percy Weasley 2 2 3
## Fleur Delacour 2 2 2
## Viktor Krum 2 2 3
## Olympe Maxime 2 2 3
## Bill Weasley 2 2 2
## Aberforth Dumbledore 2 2 3
## Dobby 2 2 2
## Petunia Dursley 1 2 3
## Vernon Dursley 2 2 3
## Nicolas Flamel 2 2 3
## Cornelius Fudge 2 2 3
## Rubeus Hagrid 2 1 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 2 3
## Charlie Weasley 2 2 2
## Dudley Dursley 2 2 3
## Luna Lovegood 2 2 3
## Fluffy 2 2 3
## Aragog 2 2 3
## Grawp 2 2 3
## Alice Longbottom 3 3 3
## Frank Longbottom 2 2 2
## Xenophilius Lovegood 3 3 4
## Hedwig 2 2 3
## Moaning Myrtle 2 2 3
## Tom Riddle Sr. 2 2 2
## Mary Riddle 2 2 2
## Arthur Weasley Fred Weasley George Weasley
## Regulus Arcturus Black 2 2 2
## Sirius Black 1 1 1
## Vincent Crabbe Sr. 2 2 2
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 2 1 1
## Albus Dumbledore 1 1 1
## Argus Filch 2 1 1
## Hermione Granger 1 1 1
## Remus Lupin 2 1 1
## Lucius Malfoy 1 2 2
## Narcissa Malfoy 2 2 2
## Minerva McGonagall 2 2 2
## Alastor Moody 1 2 2
## Peter Pettigrew 2 1 1
## Harry Potter 1 1 1
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 1 1
## Nymphadora Tonks 2 2 2
## Arthur Weasley 0 1 1
## Fred Weasley 1 0 1
## George Weasley 1 1 0
## Ginny Weasley 1 1 1
## Molly Weasley 1 1 1
## Ron Weasley 1 1 1
## Lavender Brown 2 2 2
## Seamus Finnigan 2 2 2
## Neville Longbottom 2 2 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 2 2
## Bartemius Crouch Sr. 2 2 2
## Goyle Sr. 3 2 2
## Gregory Goyle 2 2 2
## Draco Malfoy 2 2 2
## Dolores Janes Umbridge 2 2 2
## Bartemius Crouch Jr. 2 2 2
## Igor Karkaroff 3 2 2
## Percy Weasley 1 1 1
## Fleur Delacour 2 2 2
## Viktor Krum 2 2 2
## Olympe Maxime 2 2 2
## Bill Weasley 1 1 1
## Aberforth Dumbledore 2 2 2
## Dobby 2 2 2
## Petunia Dursley 2 2 2
## Vernon Dursley 2 2 2
## Nicolas Flamel 2 2 2
## Cornelius Fudge 1 2 2
## Rubeus Hagrid 2 2 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 2 2
## Charlie Weasley 1 1 1
## Dudley Dursley 2 2 2
## Luna Lovegood 2 2 2
## Fluffy 2 2 2
## Aragog 2 2 2
## Grawp 2 2 2
## Alice Longbottom 3 3 3
## Frank Longbottom 3 2 2
## Xenophilius Lovegood 3 3 3
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 3 2 2
## Mary Riddle 3 2 2
## Ginny Weasley Molly Weasley Ron Weasley
## Regulus Arcturus Black 2 2 2
## Sirius Black 1 1 1
## Vincent Crabbe Sr. 2 2 2
## Bellatrix Lestrange 2 1 1
## Lord Voldemort 1 1 1
## Albus Dumbledore 1 1 1
## Argus Filch 2 2 1
## Hermione Granger 1 1 1
## Remus Lupin 1 2 1
## Lucius Malfoy 2 1 1
## Narcissa Malfoy 2 2 2
## Minerva McGonagall 2 2 1
## Alastor Moody 2 1 1
## Peter Pettigrew 1 2 1
## Harry Potter 1 1 1
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 1 2 1
## Nymphadora Tonks 2 2 2
## Arthur Weasley 1 1 1
## Fred Weasley 1 1 1
## George Weasley 1 1 1
## Ginny Weasley 0 1 1
## Molly Weasley 1 0 1
## Ron Weasley 1 1 0
## Lavender Brown 2 2 1
## Seamus Finnigan 1 2 1
## Neville Longbottom 2 2 1
## Cho Chang 2 2 1
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 2 1
## Bartemius Crouch Sr. 2 2 2
## Goyle Sr. 2 2 2
## Gregory Goyle 2 2 1
## Draco Malfoy 2 2 1
## Dolores Janes Umbridge 2 2 1
## Bartemius Crouch Jr. 2 2 2
## Igor Karkaroff 2 2 2
## Percy Weasley 1 1 1
## Fleur Delacour 1 1 1
## Viktor Krum 2 2 2
## Olympe Maxime 2 2 2
## Bill Weasley 1 1 1
## Aberforth Dumbledore 2 2 1
## Dobby 2 2 1
## Petunia Dursley 2 2 2
## Vernon Dursley 2 2 2
## Nicolas Flamel 2 2 2
## Cornelius Fudge 2 2 2
## Rubeus Hagrid 2 2 1
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 2 2
## Charlie Weasley 1 1 1
## Dudley Dursley 2 2 2
## Luna Lovegood 2 2 1
## Fluffy 2 2 1
## Aragog 2 2 1
## Grawp 2 2 1
## Alice Longbottom 3 3 2
## Frank Longbottom 2 2 2
## Xenophilius Lovegood 3 3 2
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 2 2 2
## Mary Riddle 2 2 2
## Lavender Brown Seamus Finnigan Neville Longbottom
## Regulus Arcturus Black 3 3 2
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 3 3 2
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 2 2 1
## Albus Dumbledore 2 2 2
## Argus Filch 2 2 2
## Hermione Granger 1 1 1
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 2
## Narcissa Malfoy 3 3 2
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 2 1 1
## James Potter 3 2 2
## Lily Potter 3 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 3 3 2
## Arthur Weasley 2 2 2
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 2 1 2
## Molly Weasley 2 2 2
## Ron Weasley 1 1 1
## Lavender Brown 0 1 1
## Seamus Finnigan 1 0 1
## Neville Longbottom 1 1 0
## Cho Chang 2 1 1
## Cedric Diggory 3 2 2
## Vincent Crabbe 2 2 2
## Bartemius Crouch Sr. 3 3 2
## Goyle Sr. 3 3 2
## Gregory Goyle 2 2 2
## Draco Malfoy 2 2 1
## Dolores Janes Umbridge 2 2 2
## Bartemius Crouch Jr. 2 2 1
## Igor Karkaroff 3 3 2
## Percy Weasley 2 2 2
## Fleur Delacour 2 2 1
## Viktor Krum 2 2 2
## Olympe Maxime 3 3 2
## Bill Weasley 2 2 2
## Aberforth Dumbledore 2 2 2
## Dobby 2 2 2
## Petunia Dursley 3 2 2
## Vernon Dursley 3 2 2
## Nicolas Flamel 3 3 3
## Cornelius Fudge 3 2 2
## Rubeus Hagrid 2 2 1
## Quirinus Quirrell 2 2 2
## Rita Skeeter 3 2 2
## Charlie Weasley 2 2 2
## Dudley Dursley 3 2 2
## Luna Lovegood 2 1 1
## Fluffy 2 2 2
## Aragog 2 2 2
## Grawp 2 2 2
## Alice Longbottom 2 2 1
## Frank Longbottom 2 2 1
## Xenophilius Lovegood 3 2 2
## Hedwig 3 2 2
## Moaning Myrtle 3 2 2
## Tom Riddle Sr. 3 3 2
## Mary Riddle 3 3 2
## Cho Chang Cedric Diggory Vincent Crabbe
## Regulus Arcturus Black 2 2 2
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 2 2 1
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 1 1 2
## Albus Dumbledore 2 2 2
## Argus Filch 2 2 2
## Hermione Granger 1 2 1
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 2
## Narcissa Malfoy 2 2 2
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 1 1 1
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 2 2 3
## Arthur Weasley 2 2 2
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 2 2 2
## Molly Weasley 2 2 2
## Ron Weasley 1 2 1
## Lavender Brown 2 3 2
## Seamus Finnigan 1 2 2
## Neville Longbottom 1 2 2
## Cho Chang 0 1 2
## Cedric Diggory 1 0 2
## Vincent Crabbe 2 2 0
## Bartemius Crouch Sr. 3 3 2
## Goyle Sr. 2 2 2
## Gregory Goyle 2 2 1
## Draco Malfoy 2 2 1
## Dolores Janes Umbridge 2 2 1
## Bartemius Crouch Jr. 2 2 2
## Igor Karkaroff 2 2 3
## Percy Weasley 2 3 2
## Fleur Delacour 2 2 2
## Viktor Krum 2 1 2
## Olympe Maxime 3 3 3
## Bill Weasley 2 2 2
## Aberforth Dumbledore 2 2 2
## Dobby 2 2 2
## Petunia Dursley 2 2 2
## Vernon Dursley 2 2 2
## Nicolas Flamel 3 3 3
## Cornelius Fudge 2 2 2
## Rubeus Hagrid 2 2 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 2 2
## Charlie Weasley 2 2 2
## Dudley Dursley 2 2 2
## Luna Lovegood 2 2 2
## Fluffy 2 2 2
## Aragog 2 2 2
## Grawp 2 2 2
## Alice Longbottom 2 3 3
## Frank Longbottom 2 2 3
## Xenophilius Lovegood 3 3 3
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 2 2 3
## Mary Riddle 2 2 3
## Bartemius Crouch Sr. Goyle Sr. Gregory Goyle
## Regulus Arcturus Black 2 2 2
## Sirius Black 3 2 2
## Vincent Crabbe Sr. 1 1 2
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 2 1 1
## Albus Dumbledore 2 2 2
## Argus Filch 3 3 2
## Hermione Granger 2 2 1
## Remus Lupin 3 2 2
## Lucius Malfoy 2 2 2
## Narcissa Malfoy 3 2 2
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 2 2 1
## James Potter 3 2 2
## Lily Potter 3 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 3 2 2
## Arthur Weasley 2 3 2
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 2 2 2
## Molly Weasley 2 2 2
## Ron Weasley 2 2 1
## Lavender Brown 3 3 2
## Seamus Finnigan 3 3 2
## Neville Longbottom 2 2 2
## Cho Chang 3 2 2
## Cedric Diggory 3 2 2
## Vincent Crabbe 2 2 1
## Bartemius Crouch Sr. 0 2 3
## Goyle Sr. 2 0 1
## Gregory Goyle 3 1 0
## Draco Malfoy 3 2 1
## Dolores Janes Umbridge 3 3 2
## Bartemius Crouch Jr. 1 2 2
## Igor Karkaroff 1 2 2
## Percy Weasley 1 3 2
## Fleur Delacour 3 3 2
## Viktor Krum 2 3 2
## Olympe Maxime 2 3 3
## Bill Weasley 2 2 2
## Aberforth Dumbledore 3 3 2
## Dobby 3 3 2
## Petunia Dursley 3 3 2
## Vernon Dursley 3 3 2
## Nicolas Flamel 3 3 3
## Cornelius Fudge 2 3 2
## Rubeus Hagrid 3 2 2
## Quirinus Quirrell 3 2 2
## Rita Skeeter 3 3 2
## Charlie Weasley 2 2 2
## Dudley Dursley 3 3 2
## Luna Lovegood 3 3 2
## Fluffy 3 3 2
## Aragog 3 3 2
## Grawp 3 3 2
## Alice Longbottom 3 3 3
## Frank Longbottom 3 2 2
## Xenophilius Lovegood 4 4 3
## Hedwig 3 3 2
## Moaning Myrtle 3 3 2
## Tom Riddle Sr. 3 2 2
## Mary Riddle 3 2 2
## Draco Malfoy Dolores Janes Umbridge
## Regulus Arcturus Black 3 3
## Sirius Black 2 2
## Vincent Crabbe Sr. 2 2
## Bellatrix Lestrange 2 2
## Lord Voldemort 2 2
## Albus Dumbledore 2 1
## Argus Filch 2 1
## Hermione Granger 1 1
## Remus Lupin 2 2
## Lucius Malfoy 1 2
## Narcissa Malfoy 1 3
## Minerva McGonagall 2 1
## Alastor Moody 2 2
## Peter Pettigrew 2 2
## Harry Potter 1 1
## James Potter 2 2
## Lily Potter 2 2
## Severus Snape 2 2
## Nymphadora Tonks 3 3
## Arthur Weasley 2 2
## Fred Weasley 2 2
## George Weasley 2 2
## Ginny Weasley 2 2
## Molly Weasley 2 2
## Ron Weasley 1 1
## Lavender Brown 2 2
## Seamus Finnigan 2 2
## Neville Longbottom 1 2
## Cho Chang 2 2
## Cedric Diggory 2 2
## Vincent Crabbe 1 1
## Bartemius Crouch Sr. 3 3
## Goyle Sr. 2 3
## Gregory Goyle 1 2
## Draco Malfoy 0 2
## Dolores Janes Umbridge 2 0
## Bartemius Crouch Jr. 2 2
## Igor Karkaroff 3 3
## Percy Weasley 2 2
## Fleur Delacour 2 2
## Viktor Krum 2 2
## Olympe Maxime 3 2
## Bill Weasley 2 2
## Aberforth Dumbledore 2 2
## Dobby 1 2
## Petunia Dursley 2 2
## Vernon Dursley 2 2
## Nicolas Flamel 3 2
## Cornelius Fudge 2 1
## Rubeus Hagrid 2 2
## Quirinus Quirrell 2 2
## Rita Skeeter 2 2
## Charlie Weasley 2 2
## Dudley Dursley 2 2
## Luna Lovegood 2 2
## Fluffy 2 2
## Aragog 2 2
## Grawp 2 2
## Alice Longbottom 2 3
## Frank Longbottom 2 3
## Xenophilius Lovegood 3 3
## Hedwig 2 2
## Moaning Myrtle 2 2
## Tom Riddle Sr. 3 3
## Mary Riddle 3 3
## Bartemius Crouch Jr. Igor Karkaroff Percy Weasley
## Regulus Arcturus Black 2 2 3
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 2 2 2
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 1 1 2
## Albus Dumbledore 2 2 1
## Argus Filch 2 3 2
## Hermione Granger 1 2 2
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 2
## Narcissa Malfoy 2 2 3
## Minerva McGonagall 1 2 2
## Alastor Moody 1 2 2
## Peter Pettigrew 1 2 2
## Harry Potter 1 2 2
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 1 2 2
## Nymphadora Tonks 2 2 3
## Arthur Weasley 2 3 1
## Fred Weasley 2 2 1
## George Weasley 2 2 1
## Ginny Weasley 2 2 1
## Molly Weasley 2 2 1
## Ron Weasley 2 2 1
## Lavender Brown 2 3 2
## Seamus Finnigan 2 3 2
## Neville Longbottom 1 2 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 3
## Vincent Crabbe 2 3 2
## Bartemius Crouch Sr. 1 1 1
## Goyle Sr. 2 2 3
## Gregory Goyle 2 2 2
## Draco Malfoy 2 3 2
## Dolores Janes Umbridge 2 3 2
## Bartemius Crouch Jr. 0 2 2
## Igor Karkaroff 2 0 2
## Percy Weasley 2 2 0
## Fleur Delacour 2 2 2
## Viktor Krum 2 1 3
## Olympe Maxime 3 1 2
## Bill Weasley 2 2 1
## Aberforth Dumbledore 2 3 2
## Dobby 2 3 2
## Petunia Dursley 2 3 2
## Vernon Dursley 2 3 2
## Nicolas Flamel 3 3 2
## Cornelius Fudge 2 3 1
## Rubeus Hagrid 2 2 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 3 2
## Charlie Weasley 2 2 1
## Dudley Dursley 2 3 3
## Luna Lovegood 2 3 2
## Fluffy 2 3 2
## Aragog 2 3 2
## Grawp 2 3 2
## Alice Longbottom 2 3 3
## Frank Longbottom 2 2 3
## Xenophilius Lovegood 3 4 3
## Hedwig 2 3 3
## Moaning Myrtle 2 3 3
## Tom Riddle Sr. 2 2 3
## Mary Riddle 2 2 3
## Fleur Delacour Viktor Krum Olympe Maxime
## Regulus Arcturus Black 2 3 3
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 2 3 3
## Bellatrix Lestrange 1 2 2
## Lord Voldemort 2 2 2
## Albus Dumbledore 2 2 1
## Argus Filch 2 2 2
## Hermione Granger 1 1 2
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 3
## Narcissa Malfoy 2 3 3
## Minerva McGonagall 2 2 2
## Alastor Moody 1 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 1 1 2
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 2 3 3
## Arthur Weasley 2 2 2
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 1 2 2
## Molly Weasley 1 2 2
## Ron Weasley 1 2 2
## Lavender Brown 2 2 3
## Seamus Finnigan 2 2 3
## Neville Longbottom 1 2 2
## Cho Chang 2 2 3
## Cedric Diggory 2 1 3
## Vincent Crabbe 2 2 3
## Bartemius Crouch Sr. 3 2 2
## Goyle Sr. 3 3 3
## Gregory Goyle 2 2 3
## Draco Malfoy 2 2 3
## Dolores Janes Umbridge 2 2 2
## Bartemius Crouch Jr. 2 2 3
## Igor Karkaroff 2 1 1
## Percy Weasley 2 3 2
## Fleur Delacour 0 1 1
## Viktor Krum 1 0 2
## Olympe Maxime 1 2 0
## Bill Weasley 1 2 2
## Aberforth Dumbledore 2 2 2
## Dobby 2 2 3
## Petunia Dursley 2 2 2
## Vernon Dursley 2 2 2
## Nicolas Flamel 3 3 2
## Cornelius Fudge 2 2 2
## Rubeus Hagrid 2 2 1
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 2 2
## Charlie Weasley 2 2 2
## Dudley Dursley 2 2 2
## Luna Lovegood 2 2 3
## Fluffy 2 2 2
## Aragog 2 2 2
## Grawp 2 2 2
## Alice Longbottom 2 3 3
## Frank Longbottom 2 3 3
## Xenophilius Lovegood 3 3 4
## Hedwig 2 2 3
## Moaning Myrtle 2 2 3
## Tom Riddle Sr. 3 3 3
## Mary Riddle 3 3 3
## Bill Weasley Aberforth Dumbledore Dobby
## Regulus Arcturus Black 2 3 2
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 2 3 2
## Bellatrix Lestrange 2 2 1
## Lord Voldemort 1 2 2
## Albus Dumbledore 1 1 2
## Argus Filch 2 2 2
## Hermione Granger 1 1 1
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 1
## Narcissa Malfoy 2 3 2
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 1 1 1
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 2 3 2
## Arthur Weasley 1 2 2
## Fred Weasley 1 2 2
## George Weasley 1 2 2
## Ginny Weasley 1 2 2
## Molly Weasley 1 2 2
## Ron Weasley 1 1 1
## Lavender Brown 2 2 2
## Seamus Finnigan 2 2 2
## Neville Longbottom 2 2 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 2 2
## Bartemius Crouch Sr. 2 3 3
## Goyle Sr. 2 3 3
## Gregory Goyle 2 2 2
## Draco Malfoy 2 2 1
## Dolores Janes Umbridge 2 2 2
## Bartemius Crouch Jr. 2 2 2
## Igor Karkaroff 2 3 3
## Percy Weasley 1 2 2
## Fleur Delacour 1 2 2
## Viktor Krum 2 2 2
## Olympe Maxime 2 2 3
## Bill Weasley 0 2 2
## Aberforth Dumbledore 2 0 1
## Dobby 2 1 0
## Petunia Dursley 2 2 2
## Vernon Dursley 2 2 2
## Nicolas Flamel 2 2 3
## Cornelius Fudge 2 2 2
## Rubeus Hagrid 2 2 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 2 2
## Charlie Weasley 1 2 2
## Dudley Dursley 2 2 2
## Luna Lovegood 2 2 2
## Fluffy 2 2 2
## Aragog 2 2 2
## Grawp 2 2 2
## Alice Longbottom 3 3 3
## Frank Longbottom 2 3 3
## Xenophilius Lovegood 3 3 3
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 2 3 3
## Mary Riddle 2 3 3
## Petunia Dursley Vernon Dursley Nicolas Flamel
## Regulus Arcturus Black 3 3 3
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 3 3 3
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 2 2 2
## Albus Dumbledore 1 1 1
## Argus Filch 2 2 2
## Hermione Granger 2 2 2
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 3
## Narcissa Malfoy 3 3 3
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 1 1 2
## James Potter 2 2 2
## Lily Potter 1 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 3 3 3
## Arthur Weasley 2 2 2
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 2 2 2
## Molly Weasley 2 2 2
## Ron Weasley 2 2 2
## Lavender Brown 3 3 3
## Seamus Finnigan 2 2 3
## Neville Longbottom 2 2 3
## Cho Chang 2 2 3
## Cedric Diggory 2 2 3
## Vincent Crabbe 2 2 3
## Bartemius Crouch Sr. 3 3 3
## Goyle Sr. 3 3 3
## Gregory Goyle 2 2 3
## Draco Malfoy 2 2 3
## Dolores Janes Umbridge 2 2 2
## Bartemius Crouch Jr. 2 2 3
## Igor Karkaroff 3 3 3
## Percy Weasley 2 2 2
## Fleur Delacour 2 2 3
## Viktor Krum 2 2 3
## Olympe Maxime 2 2 2
## Bill Weasley 2 2 2
## Aberforth Dumbledore 2 2 2
## Dobby 2 2 3
## Petunia Dursley 0 1 2
## Vernon Dursley 1 0 2
## Nicolas Flamel 2 2 0
## Cornelius Fudge 2 2 2
## Rubeus Hagrid 2 2 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 2 2
## Charlie Weasley 2 2 2
## Dudley Dursley 1 1 3
## Luna Lovegood 2 2 3
## Fluffy 2 2 3
## Aragog 2 2 3
## Grawp 2 2 3
## Alice Longbottom 3 3 4
## Frank Longbottom 3 3 3
## Xenophilius Lovegood 3 3 4
## Hedwig 2 2 3
## Moaning Myrtle 2 2 3
## Tom Riddle Sr. 3 3 3
## Mary Riddle 3 3 3
## Cornelius Fudge Rubeus Hagrid Quirinus Quirrell
## Regulus Arcturus Black 3 2 2
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 2 2 2
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 2 1 1
## Albus Dumbledore 1 1 1
## Argus Filch 2 1 2
## Hermione Granger 2 1 1
## Remus Lupin 2 2 2
## Lucius Malfoy 1 2 2
## Narcissa Malfoy 2 2 2
## Minerva McGonagall 2 1 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 1 1 1
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 1 2
## Nymphadora Tonks 3 2 2
## Arthur Weasley 1 2 2
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 2 2 2
## Molly Weasley 2 2 2
## Ron Weasley 2 1 2
## Lavender Brown 3 2 2
## Seamus Finnigan 2 2 2
## Neville Longbottom 2 1 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 2 2
## Bartemius Crouch Sr. 2 3 3
## Goyle Sr. 3 2 2
## Gregory Goyle 2 2 2
## Draco Malfoy 2 2 2
## Dolores Janes Umbridge 1 2 2
## Bartemius Crouch Jr. 2 2 2
## Igor Karkaroff 3 2 2
## Percy Weasley 1 2 2
## Fleur Delacour 2 2 2
## Viktor Krum 2 2 2
## Olympe Maxime 2 1 2
## Bill Weasley 2 2 2
## Aberforth Dumbledore 2 2 2
## Dobby 2 2 2
## Petunia Dursley 2 2 2
## Vernon Dursley 2 2 2
## Nicolas Flamel 2 2 2
## Cornelius Fudge 0 2 2
## Rubeus Hagrid 2 0 2
## Quirinus Quirrell 2 2 0
## Rita Skeeter 2 2 2
## Charlie Weasley 2 2 2
## Dudley Dursley 2 1 2
## Luna Lovegood 2 2 2
## Fluffy 2 1 2
## Aragog 2 1 2
## Grawp 2 1 2
## Alice Longbottom 3 2 3
## Frank Longbottom 3 2 2
## Xenophilius Lovegood 3 3 3
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 3 2 2
## Mary Riddle 3 2 2
## Rita Skeeter Charlie Weasley Dudley Dursley
## Regulus Arcturus Black 3 2 3
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 3 2 3
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 2 1 2
## Albus Dumbledore 1 1 2
## Argus Filch 2 2 2
## Hermione Granger 2 1 2
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 2
## Narcissa Malfoy 3 2 3
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 1 1 1
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 3 2 3
## Arthur Weasley 2 1 2
## Fred Weasley 2 1 2
## George Weasley 2 1 2
## Ginny Weasley 2 1 2
## Molly Weasley 2 1 2
## Ron Weasley 2 1 2
## Lavender Brown 3 2 3
## Seamus Finnigan 2 2 2
## Neville Longbottom 2 2 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 2 2
## Bartemius Crouch Sr. 3 2 3
## Goyle Sr. 3 2 3
## Gregory Goyle 2 2 2
## Draco Malfoy 2 2 2
## Dolores Janes Umbridge 2 2 2
## Bartemius Crouch Jr. 2 2 2
## Igor Karkaroff 3 2 3
## Percy Weasley 2 1 3
## Fleur Delacour 2 2 2
## Viktor Krum 2 2 2
## Olympe Maxime 2 2 2
## Bill Weasley 2 1 2
## Aberforth Dumbledore 2 2 2
## Dobby 2 2 2
## Petunia Dursley 2 2 1
## Vernon Dursley 2 2 1
## Nicolas Flamel 2 2 3
## Cornelius Fudge 2 2 2
## Rubeus Hagrid 2 2 1
## Quirinus Quirrell 2 2 2
## Rita Skeeter 0 2 2
## Charlie Weasley 2 0 2
## Dudley Dursley 2 2 0
## Luna Lovegood 2 2 2
## Fluffy 2 2 2
## Aragog 2 2 2
## Grawp 2 2 2
## Alice Longbottom 3 3 3
## Frank Longbottom 3 2 3
## Xenophilius Lovegood 3 3 3
## Hedwig 2 2 2
## Moaning Myrtle 2 2 2
## Tom Riddle Sr. 3 2 3
## Mary Riddle 3 2 3
## Luna Lovegood Fluffy Aragog Grawp Alice Longbottom
## Regulus Arcturus Black 3 3 3 3 3
## Sirius Black 2 2 2 2 3
## Vincent Crabbe Sr. 3 3 3 3 3
## Bellatrix Lestrange 2 2 2 2 3
## Lord Voldemort 2 2 2 2 2
## Albus Dumbledore 2 2 2 2 3
## Argus Filch 2 2 2 2 3
## Hermione Granger 1 1 1 1 2
## Remus Lupin 2 2 2 2 3
## Lucius Malfoy 2 2 2 2 3
## Narcissa Malfoy 3 3 3 3 3
## Minerva McGonagall 2 2 2 2 3
## Alastor Moody 2 2 2 2 3
## Peter Pettigrew 2 2 2 2 3
## Harry Potter 1 1 1 1 2
## James Potter 2 2 2 2 3
## Lily Potter 2 2 2 2 3
## Severus Snape 2 2 2 2 3
## Nymphadora Tonks 3 3 3 3 3
## Arthur Weasley 2 2 2 2 3
## Fred Weasley 2 2 2 2 3
## George Weasley 2 2 2 2 3
## Ginny Weasley 2 2 2 2 3
## Molly Weasley 2 2 2 2 3
## Ron Weasley 1 1 1 1 2
## Lavender Brown 2 2 2 2 2
## Seamus Finnigan 1 2 2 2 2
## Neville Longbottom 1 2 2 2 1
## Cho Chang 2 2 2 2 2
## Cedric Diggory 2 2 2 2 3
## Vincent Crabbe 2 2 2 2 3
## Bartemius Crouch Sr. 3 3 3 3 3
## Goyle Sr. 3 3 3 3 3
## Gregory Goyle 2 2 2 2 3
## Draco Malfoy 2 2 2 2 2
## Dolores Janes Umbridge 2 2 2 2 3
## Bartemius Crouch Jr. 2 2 2 2 2
## Igor Karkaroff 3 3 3 3 3
## Percy Weasley 2 2 2 2 3
## Fleur Delacour 2 2 2 2 2
## Viktor Krum 2 2 2 2 3
## Olympe Maxime 3 2 2 2 3
## Bill Weasley 2 2 2 2 3
## Aberforth Dumbledore 2 2 2 2 3
## Dobby 2 2 2 2 3
## Petunia Dursley 2 2 2 2 3
## Vernon Dursley 2 2 2 2 3
## Nicolas Flamel 3 3 3 3 4
## Cornelius Fudge 2 2 2 2 3
## Rubeus Hagrid 2 1 1 1 2
## Quirinus Quirrell 2 2 2 2 3
## Rita Skeeter 2 2 2 2 3
## Charlie Weasley 2 2 2 2 3
## Dudley Dursley 2 2 2 2 3
## Luna Lovegood 0 2 2 2 2
## Fluffy 2 0 2 2 3
## Aragog 2 2 0 2 3
## Grawp 2 2 2 0 3
## Alice Longbottom 2 3 3 3 0
## Frank Longbottom 2 3 3 3 1
## Xenophilius Lovegood 1 3 3 3 3
## Hedwig 2 2 2 2 3
## Moaning Myrtle 2 2 2 2 3
## Tom Riddle Sr. 3 3 3 3 3
## Mary Riddle 3 3 3 3 3
## Frank Longbottom Xenophilius Lovegood Hedwig
## Regulus Arcturus Black 2 4 3
## Sirius Black 2 3 2
## Vincent Crabbe Sr. 2 4 3
## Bellatrix Lestrange 2 3 2
## Lord Voldemort 1 3 2
## Albus Dumbledore 2 3 2
## Argus Filch 3 3 2
## Hermione Granger 2 2 2
## Remus Lupin 2 3 2
## Lucius Malfoy 2 3 2
## Narcissa Malfoy 2 4 3
## Minerva McGonagall 2 3 2
## Alastor Moody 2 3 2
## Peter Pettigrew 2 3 2
## Harry Potter 2 2 1
## James Potter 2 3 2
## Lily Potter 2 3 2
## Severus Snape 2 3 2
## Nymphadora Tonks 2 4 3
## Arthur Weasley 3 3 2
## Fred Weasley 2 3 2
## George Weasley 2 3 2
## Ginny Weasley 2 3 2
## Molly Weasley 2 3 2
## Ron Weasley 2 2 2
## Lavender Brown 2 3 3
## Seamus Finnigan 2 2 2
## Neville Longbottom 1 2 2
## Cho Chang 2 3 2
## Cedric Diggory 2 3 2
## Vincent Crabbe 3 3 2
## Bartemius Crouch Sr. 3 4 3
## Goyle Sr. 2 4 3
## Gregory Goyle 2 3 2
## Draco Malfoy 2 3 2
## Dolores Janes Umbridge 3 3 2
## Bartemius Crouch Jr. 2 3 2
## Igor Karkaroff 2 4 3
## Percy Weasley 3 3 3
## Fleur Delacour 2 3 2
## Viktor Krum 3 3 2
## Olympe Maxime 3 4 3
## Bill Weasley 2 3 2
## Aberforth Dumbledore 3 3 2
## Dobby 3 3 2
## Petunia Dursley 3 3 2
## Vernon Dursley 3 3 2
## Nicolas Flamel 3 4 3
## Cornelius Fudge 3 3 2
## Rubeus Hagrid 2 3 2
## Quirinus Quirrell 2 3 2
## Rita Skeeter 3 3 2
## Charlie Weasley 2 3 2
## Dudley Dursley 3 3 2
## Luna Lovegood 2 1 2
## Fluffy 3 3 2
## Aragog 3 3 2
## Grawp 3 3 2
## Alice Longbottom 1 3 3
## Frank Longbottom 0 3 3
## Xenophilius Lovegood 3 0 3
## Hedwig 3 3 0
## Moaning Myrtle 3 3 2
## Tom Riddle Sr. 2 4 3
## Mary Riddle 2 4 3
## Moaning Myrtle Tom Riddle Sr. Mary Riddle
## Regulus Arcturus Black 3 2 2
## Sirius Black 2 2 2
## Vincent Crabbe Sr. 3 2 2
## Bellatrix Lestrange 2 2 2
## Lord Voldemort 2 1 1
## Albus Dumbledore 2 2 2
## Argus Filch 2 3 3
## Hermione Granger 2 2 2
## Remus Lupin 2 2 2
## Lucius Malfoy 2 2 2
## Narcissa Malfoy 3 2 2
## Minerva McGonagall 2 2 2
## Alastor Moody 2 2 2
## Peter Pettigrew 2 2 2
## Harry Potter 1 2 2
## James Potter 2 2 2
## Lily Potter 2 2 2
## Severus Snape 2 2 2
## Nymphadora Tonks 3 2 2
## Arthur Weasley 2 3 3
## Fred Weasley 2 2 2
## George Weasley 2 2 2
## Ginny Weasley 2 2 2
## Molly Weasley 2 2 2
## Ron Weasley 2 2 2
## Lavender Brown 3 3 3
## Seamus Finnigan 2 3 3
## Neville Longbottom 2 2 2
## Cho Chang 2 2 2
## Cedric Diggory 2 2 2
## Vincent Crabbe 2 3 3
## Bartemius Crouch Sr. 3 3 3
## Goyle Sr. 3 2 2
## Gregory Goyle 2 2 2
## Draco Malfoy 2 3 3
## Dolores Janes Umbridge 2 3 3
## Bartemius Crouch Jr. 2 2 2
## Igor Karkaroff 3 2 2
## Percy Weasley 3 3 3
## Fleur Delacour 2 3 3
## Viktor Krum 2 3 3
## Olympe Maxime 3 3 3
## Bill Weasley 2 2 2
## Aberforth Dumbledore 2 3 3
## Dobby 2 3 3
## Petunia Dursley 2 3 3
## Vernon Dursley 2 3 3
## Nicolas Flamel 3 3 3
## Cornelius Fudge 2 3 3
## Rubeus Hagrid 2 2 2
## Quirinus Quirrell 2 2 2
## Rita Skeeter 2 3 3
## Charlie Weasley 2 2 2
## Dudley Dursley 2 3 3
## Luna Lovegood 2 3 3
## Fluffy 2 3 3
## Aragog 2 3 3
## Grawp 2 3 3
## Alice Longbottom 3 3 3
## Frank Longbottom 3 2 2
## Xenophilius Lovegood 3 4 4
## Hedwig 2 3 3
## Moaning Myrtle 0 3 3
## Tom Riddle Sr. 3 0 1
## Mary Riddle 3 1 0Network Diameter
In contrast to the shortest path between two nodes, we can also calculate the longest path, or diameter. The network diameter describes the longest possible distance between two vertices in terms of the most efficient path.
In our network, the longest path connects 4 edges (5 characters).
get_diameter returns a path with the actual diameter. In our case, the diameter passes by: Regulus Arcturus Black, Sirius Black, Hermione Granger, Luna Lovegood and Xenophilius Lovegood. From this result, we can assume that some of these characters have high centrality measures (to confirm below).
diameter_characters<-diameter(g11, directed=F)
diameter_characters #length of diameter## [1] 4get_diameter(g11) #list of characters that are part of the diameter## + 5/65 vertices, named, from 861146a:
## [1] Regulus Arcturus Black Sirius Black Hermione Granger
## [4] Luna Lovegood Xenophilius LovegoodPlotting Longest Path in Harry Potter Network
The following graph plots the whole network but highlights the diameter in red. Surprisingly, the diameter goes through Hermione Granger, instead of Harry Potter. Harry Potter seems to be able to make the same connections as Hermione Granger, but wasn’t chosen, most likely, because he has a smaller level of degree (discussed later).
par(mar=c(1,1,3,1)) #layout of the RMarkdown
g11_diameter <- g11
node_diameter <- get.diameter(g11_diameter, directed = FALSE) #assignment the diameter parth to the igraph
#Specifying the nodes' and edge's characteristics (color, size, transparency, width)
V(g11_diameter)$color <- scales::alpha(V(g11_diameter)$color, alpha = 4)## Warning in length(vattrs[[name]]) <- vc: length of NULL cannot be changedV(g11_diameter)$size <- 10
V(g11_diameter)[node_diameter]$color <- "red"
V(g11_diameter)[node_diameter]$size <- 20
E(g11_diameter)$color <- "grey"
E(g11_diameter)$width <- 0.01
E(g11_diameter, path = node_diameter)$color <- "red"
E(g11_diameter, path = node_diameter)$width <- 3
#Plotting the whole graph
plot(g11_diameter,
layout = mylayout,
legend = TRUE,
vertex.label = V(g11_diameter)$Character,
vertex.frame.color = "gray",
vertex.label.color = "black",
vertex.label.cex = 0.6,
vertex.label.dist=3,
edge.lty = E(g11_diameter)$lty,
asp = 0.75,
rescale = FALSE,
ylim = c(-4.5,4),
xlim = c(-5,2))
legend("topleft", legend = "", cex = 1.25, bty = "n", ncol = 1,
title = "The Network's Diameter")
Giant Component
In most of the research topics for network analysis, network features are related to the largest connected component of a graph (Newman 2010). In order to get that for an igraph or a network object, giant_component_extract function is specified. This function extracts the strongest components of the input network as igraph objects.
In the end, for this network, the function only gives us 2 lists of the edges. This may be a consequence of a small node network, where all the edges are strong compenents (due to the limited size of the network).
giant_component_extract(g11)## [[1]]
## IGRAPH 871171d UN-- 65 330 --
## + attr: name (v/c), Alignment (v/c), Gender_Male (v/n), House
## | (v/c), Color (v/c)
## + edges from 871171d (vertex names):
## [1] Regulus Arcturus Black--Sirius Black
## [2] Regulus Arcturus Black--Vincent Crabbe Sr.
## [3] Regulus Arcturus Black--Bellatrix Lestrange
## [4] Regulus Arcturus Black--Lord Voldemort
## [5] Sirius Black --Albus Dumbledore
## [6] Sirius Black --Argus Filch
## [7] Sirius Black --Hermione Granger
## + ... omitted several edges
##
## [[2]]
## [,1] [,2]
## [1,] "Regulus Arcturus Black" "Sirius Black"
## [2,] "Regulus Arcturus Black" "Vincent Crabbe Sr."
## [3,] "Regulus Arcturus Black" "Bellatrix Lestrange"
## [4,] "Regulus Arcturus Black" "Lord Voldemort"
## [5,] "Sirius Black" "Albus Dumbledore"
## [6,] "Sirius Black" "Argus Filch"
## [7,] "Sirius Black" "Hermione Granger"
## [8,] "Sirius Black" "Bellatrix Lestrange"
## [9,] "Sirius Black" "Remus Lupin"
## [10,] "Sirius Black" "Lucius Malfoy"
## [11,] "Sirius Black" "Narcissa Malfoy"
## [12,] "Sirius Black" "Minerva McGonagall"
## [13,] "Sirius Black" "Alastor Moody"
## [14,] "Sirius Black" "Peter Pettigrew"
## [15,] "Sirius Black" "Harry Potter"
## [16,] "Sirius Black" "James Potter"
## [17,] "Sirius Black" "Lily Potter"
## [18,] "Sirius Black" "Lord Voldemort"
## [19,] "Sirius Black" "Severus Snape"
## [20,] "Sirius Black" "Nymphadora Tonks"
## [21,] "Sirius Black" "Arthur Weasley"
## [22,] "Sirius Black" "Fred Weasley"
## [23,] "Sirius Black" "George Weasley"
## [24,] "Sirius Black" "Ginny Weasley"
## [25,] "Sirius Black" "Molly Weasley"
## [26,] "Sirius Black" "Ron Weasley"
## [27,] "Lavender Brown" "Seamus Finnigan"
## [28,] "Lavender Brown" "Hermione Granger"
## [29,] "Lavender Brown" "Neville Longbottom"
## [30,] "Lavender Brown" "Ron Weasley"
## [31,] "Cho Chang" "Cedric Diggory"
## [32,] "Cho Chang" "Seamus Finnigan"
## [33,] "Cho Chang" "Hermione Granger"
## [34,] "Cho Chang" "Neville Longbottom"
## [35,] "Cho Chang" "Harry Potter"
## [36,] "Cho Chang" "Lord Voldemort"
## [37,] "Cho Chang" "Ron Weasley"
## [38,] "Vincent Crabbe Sr." "Vincent Crabbe"
## [39,] "Vincent Crabbe Sr." "Bartemius Crouch Sr."
## [40,] "Vincent Crabbe Sr." "Goyle Sr."
## [41,] "Vincent Crabbe Sr." "Bellatrix Lestrange"
## [42,] "Vincent Crabbe Sr." "Lucius Malfoy"
## [43,] "Vincent Crabbe Sr." "Peter Pettigrew"
## [44,] "Vincent Crabbe Sr." "Lord Voldemort"
## [45,] "Vincent Crabbe Sr." "Severus Snape"
## [46,] "Vincent Crabbe" "Gregory Goyle"
## [47,] "Vincent Crabbe" "Hermione Granger"
## [48,] "Vincent Crabbe" "Draco Malfoy"
## [49,] "Vincent Crabbe" "Harry Potter"
## [50,] "Vincent Crabbe" "Dolores Janes Umbridge"
## [51,] "Vincent Crabbe" "Ron Weasley"
## [52,] "Bartemius Crouch Sr." "Bartemius Crouch Jr."
## [53,] "Bartemius Crouch Sr." "Igor Karkaroff"
## [54,] "Bartemius Crouch Sr." "Percy Weasley"
## [55,] "Bartemius Crouch Jr." "Hermione Granger"
## [56,] "Bartemius Crouch Jr." "Neville Longbottom"
## [57,] "Bartemius Crouch Jr." "Minerva McGonagall"
## [58,] "Bartemius Crouch Jr." "Alastor Moody"
## [59,] "Bartemius Crouch Jr." "Peter Pettigrew"
## [60,] "Bartemius Crouch Jr." "Harry Potter"
## [61,] "Bartemius Crouch Jr." "Lord Voldemort"
## [62,] "Bartemius Crouch Jr." "Severus Snape"
## [63,] "Fleur Delacour" "Hermione Granger"
## [64,] "Fleur Delacour" "Viktor Krum"
## [65,] "Fleur Delacour" "Bellatrix Lestrange"
## [66,] "Fleur Delacour" "Neville Longbottom"
## [67,] "Fleur Delacour" "Olympe Maxime"
## [68,] "Fleur Delacour" "Alastor Moody"
## [69,] "Fleur Delacour" "Harry Potter"
## [70,] "Fleur Delacour" "Bill Weasley"
## [71,] "Fleur Delacour" "Ginny Weasley"
## [72,] "Fleur Delacour" "Molly Weasley"
## [73,] "Fleur Delacour" "Ron Weasley"
## [74,] "Cedric Diggory" "Viktor Krum"
## [75,] "Cedric Diggory" "Harry Potter"
## [76,] "Cedric Diggory" "Lord Voldemort"
## [77,] "Aberforth Dumbledore" "Albus Dumbledore"
## [78,] "Aberforth Dumbledore" "Hermione Granger"
## [79,] "Aberforth Dumbledore" "Harry Potter"
## [80,] "Aberforth Dumbledore" "Ron Weasley"
## [81,] "Aberforth Dumbledore" "Dobby"
## [82,] "Albus Dumbledore" "Petunia Dursley"
## [83,] "Albus Dumbledore" "Vernon Dursley"
## [84,] "Albus Dumbledore" "Argus Filch"
## [85,] "Albus Dumbledore" "Nicolas Flamel"
## [86,] "Albus Dumbledore" "Cornelius Fudge"
## [87,] "Albus Dumbledore" "Hermione Granger"
## [88,] "Albus Dumbledore" "Rubeus Hagrid"
## [89,] "Albus Dumbledore" "Bellatrix Lestrange"
## [90,] "Albus Dumbledore" "Remus Lupin"
## [91,] "Albus Dumbledore" "Olympe Maxime"
## [92,] "Albus Dumbledore" "Minerva McGonagall"
## [93,] "Albus Dumbledore" "Alastor Moody"
## [94,] "Albus Dumbledore" "Peter Pettigrew"
## [95,] "Albus Dumbledore" "Harry Potter"
## [96,] "Albus Dumbledore" "James Potter"
## [97,] "Albus Dumbledore" "Lily Potter"
## [98,] "Albus Dumbledore" "Quirinus Quirrell"
## [99,] "Albus Dumbledore" "Lord Voldemort"
## [100,] "Albus Dumbledore" "Rita Skeeter"
## [101,] "Albus Dumbledore" "Severus Snape"
## [102,] "Albus Dumbledore" "Dolores Janes Umbridge"
## [103,] "Albus Dumbledore" "Arthur Weasley"
## [104,] "Albus Dumbledore" "Bill Weasley"
## [105,] "Albus Dumbledore" "Charlie Weasley"
## [106,] "Albus Dumbledore" "Fred Weasley"
## [107,] "Albus Dumbledore" "George Weasley"
## [108,] "Albus Dumbledore" "Ginny Weasley"
## [109,] "Albus Dumbledore" "Molly Weasley"
## [110,] "Albus Dumbledore" "Percy Weasley"
## [111,] "Albus Dumbledore" "Ron Weasley"
## [112,] "Dudley Dursley" "Petunia Dursley"
## [113,] "Dudley Dursley" "Vernon Dursley"
## [114,] "Dudley Dursley" "Rubeus Hagrid"
## [115,] "Dudley Dursley" "Harry Potter"
## [116,] "Petunia Dursley" "Vernon Dursley"
## [117,] "Petunia Dursley" "Harry Potter"
## [118,] "Petunia Dursley" "Lily Potter"
## [119,] "Vernon Dursley" "Harry Potter"
## [120,] "Argus Filch" "Hermione Granger"
## [121,] "Argus Filch" "Rubeus Hagrid"
## [122,] "Argus Filch" "Minerva McGonagall"
## [123,] "Argus Filch" "Peter Pettigrew"
## [124,] "Argus Filch" "Harry Potter"
## [125,] "Argus Filch" "James Potter"
## [126,] "Argus Filch" "Severus Snape"
## [127,] "Argus Filch" "Dolores Janes Umbridge"
## [128,] "Argus Filch" "Fred Weasley"
## [129,] "Argus Filch" "George Weasley"
## [130,] "Argus Filch" "Ron Weasley"
## [131,] "Seamus Finnigan" "Hermione Granger"
## [132,] "Seamus Finnigan" "Neville Longbottom"
## [133,] "Seamus Finnigan" "Luna Lovegood"
## [134,] "Seamus Finnigan" "Harry Potter"
## [135,] "Seamus Finnigan" "Ginny Weasley"
## [136,] "Seamus Finnigan" "Ron Weasley"
## [137,] "Cornelius Fudge" "Lucius Malfoy"
## [138,] "Cornelius Fudge" "Harry Potter"
## [139,] "Cornelius Fudge" "Dolores Janes Umbridge"
## [140,] "Cornelius Fudge" "Arthur Weasley"
## [141,] "Cornelius Fudge" "Percy Weasley"
## [142,] "Goyle Sr." "Gregory Goyle"
## [143,] "Goyle Sr." "Lord Voldemort"
## [144,] "Gregory Goyle" "Hermione Granger"
## [145,] "Gregory Goyle" "Draco Malfoy"
## [146,] "Gregory Goyle" "Harry Potter"
## [147,] "Gregory Goyle" "Lord Voldemort"
## [148,] "Gregory Goyle" "Ron Weasley"
## [149,] "Hermione Granger" "Rubeus Hagrid"
## [150,] "Hermione Granger" "Viktor Krum"
## [151,] "Hermione Granger" "Neville Longbottom"
## [152,] "Hermione Granger" "Luna Lovegood"
## [153,] "Hermione Granger" "Draco Malfoy"
## [154,] "Hermione Granger" "Minerva McGonagall"
## [155,] "Hermione Granger" "Alastor Moody"
## [156,] "Hermione Granger" "Peter Pettigrew"
## [157,] "Hermione Granger" "Harry Potter"
## [158,] "Hermione Granger" "Quirinus Quirrell"
## [159,] "Hermione Granger" "Lord Voldemort"
## [160,] "Hermione Granger" "Severus Snape"
## [161,] "Hermione Granger" "Dolores Janes Umbridge"
## [162,] "Hermione Granger" "Arthur Weasley"
## [163,] "Hermione Granger" "Bill Weasley"
## [164,] "Hermione Granger" "Charlie Weasley"
## [165,] "Hermione Granger" "Fred Weasley"
## [166,] "Hermione Granger" "George Weasley"
## [167,] "Hermione Granger" "Ginny Weasley"
## [168,] "Hermione Granger" "Molly Weasley"
## [169,] "Hermione Granger" "Ron Weasley"
## [170,] "Hermione Granger" "Dobby"
## [171,] "Hermione Granger" "Fluffy"
## [172,] "Hermione Granger" "Aragog"
## [173,] "Hermione Granger" "Grawp"
## [174,] "Rubeus Hagrid" "Neville Longbottom"
## [175,] "Rubeus Hagrid" "Olympe Maxime"
## [176,] "Rubeus Hagrid" "Minerva McGonagall"
## [177,] "Rubeus Hagrid" "Harry Potter"
## [178,] "Rubeus Hagrid" "Lord Voldemort"
## [179,] "Rubeus Hagrid" "Severus Snape"
## [180,] "Rubeus Hagrid" "Ron Weasley"
## [181,] "Rubeus Hagrid" "Fluffy"
## [182,] "Rubeus Hagrid" "Aragog"
## [183,] "Rubeus Hagrid" "Grawp"
## [184,] "Igor Karkaroff" "Viktor Krum"
## [185,] "Igor Karkaroff" "Olympe Maxime"
## [186,] "Igor Karkaroff" "Lord Voldemort"
## [187,] "Viktor Krum" "Harry Potter"
## [188,] "Bellatrix Lestrange" "Lucius Malfoy"
## [189,] "Bellatrix Lestrange" "Narcissa Malfoy"
## [190,] "Bellatrix Lestrange" "Harry Potter"
## [191,] "Bellatrix Lestrange" "Lord Voldemort"
## [192,] "Bellatrix Lestrange" "Nymphadora Tonks"
## [193,] "Bellatrix Lestrange" "Molly Weasley"
## [194,] "Bellatrix Lestrange" "Ron Weasley"
## [195,] "Bellatrix Lestrange" "Dobby"
## [196,] "Alice Longbottom" "Frank Longbottom"
## [197,] "Alice Longbottom" "Neville Longbottom"
## [198,] "Frank Longbottom" "Neville Longbottom"
## [199,] "Frank Longbottom" "Lord Voldemort"
## [200,] "Neville Longbottom" "Luna Lovegood"
## [201,] "Neville Longbottom" "Draco Malfoy"
## [202,] "Neville Longbottom" "Harry Potter"
## [203,] "Neville Longbottom" "Lord Voldemort"
## [204,] "Neville Longbottom" "Ron Weasley"
## [205,] "Luna Lovegood" "Xenophilius Lovegood"
## [206,] "Luna Lovegood" "Harry Potter"
## [207,] "Luna Lovegood" "Ron Weasley"
## [208,] "Remus Lupin" "Minerva McGonagall"
## [209,] "Remus Lupin" "Peter Pettigrew"
## [210,] "Remus Lupin" "Harry Potter"
## [211,] "Remus Lupin" "James Potter"
## [212,] "Remus Lupin" "Lord Voldemort"
## [213,] "Remus Lupin" "Nymphadora Tonks"
## [214,] "Remus Lupin" "Fred Weasley"
## [215,] "Remus Lupin" "George Weasley"
## [216,] "Remus Lupin" "Ginny Weasley"
## [217,] "Remus Lupin" "Ron Weasley"
## [218,] "Draco Malfoy" "Lucius Malfoy"
## [219,] "Draco Malfoy" "Narcissa Malfoy"
## [220,] "Draco Malfoy" "Harry Potter"
## [221,] "Draco Malfoy" "Ron Weasley"
## [222,] "Draco Malfoy" "Dobby"
## [223,] "Lucius Malfoy" "Narcissa Malfoy"
## [224,] "Lucius Malfoy" "Harry Potter"
## [225,] "Lucius Malfoy" "Lord Voldemort"
## [226,] "Lucius Malfoy" "Arthur Weasley"
## [227,] "Lucius Malfoy" "Molly Weasley"
## [228,] "Lucius Malfoy" "Ron Weasley"
## [229,] "Lucius Malfoy" "Dobby"
## [230,] "Narcissa Malfoy" "Lord Voldemort"
## [231,] "Minerva McGonagall" "Harry Potter"
## [232,] "Minerva McGonagall" "Lord Voldemort"
## [233,] "Minerva McGonagall" "Dolores Janes Umbridge"
## [234,] "Minerva McGonagall" "Ron Weasley"
## [235,] "Alastor Moody" "Harry Potter"
## [236,] "Alastor Moody" "Lord Voldemort"
## [237,] "Alastor Moody" "Arthur Weasley"
## [238,] "Alastor Moody" "Molly Weasley"
## [239,] "Alastor Moody" "Ron Weasley"
## [240,] "Peter Pettigrew" "Harry Potter"
## [241,] "Peter Pettigrew" "James Potter"
## [242,] "Peter Pettigrew" "Lily Potter"
## [243,] "Peter Pettigrew" "Lord Voldemort"
## [244,] "Peter Pettigrew" "Fred Weasley"
## [245,] "Peter Pettigrew" "George Weasley"
## [246,] "Peter Pettigrew" "Ginny Weasley"
## [247,] "Peter Pettigrew" "Ron Weasley"
## [248,] "Harry Potter" "James Potter"
## [249,] "Harry Potter" "Lily Potter"
## [250,] "Harry Potter" "Quirinus Quirrell"
## [251,] "Harry Potter" "Lord Voldemort"
## [252,] "Harry Potter" "Rita Skeeter"
## [253,] "Harry Potter" "Severus Snape"
## [254,] "Harry Potter" "Dolores Janes Umbridge"
## [255,] "Harry Potter" "Arthur Weasley"
## [256,] "Harry Potter" "Bill Weasley"
## [257,] "Harry Potter" "Charlie Weasley"
## [258,] "Harry Potter" "Fred Weasley"
## [259,] "Harry Potter" "George Weasley"
## [260,] "Harry Potter" "Ginny Weasley"
## [261,] "Harry Potter" "Molly Weasley"
## [262,] "Harry Potter" "Ron Weasley"
## [263,] "Harry Potter" "Dobby"
## [264,] "Harry Potter" "Fluffy"
## [265,] "Harry Potter" "Hedwig"
## [266,] "Harry Potter" "Moaning Myrtle"
## [267,] "Harry Potter" "Aragog"
## [268,] "Harry Potter" "Grawp"
## [269,] "James Potter" "Lily Potter"
## [270,] "James Potter" "Lord Voldemort"
## [271,] "Lily Potter" "Lord Voldemort"
## [272,] "Lily Potter" "Severus Snape"
## [273,] "Quirinus Quirrell" "Lord Voldemort"
## [274,] "Tom Riddle Sr." "Mary Riddle"
## [275,] "Tom Riddle Sr." "Lord Voldemort"
## [276,] "Mary Riddle" "Lord Voldemort"
## [277,] "Lord Voldemort" "Severus Snape"
## [278,] "Lord Voldemort" "Nymphadora Tonks"
## [279,] "Lord Voldemort" "Bill Weasley"
## [280,] "Lord Voldemort" "Charlie Weasley"
## [281,] "Lord Voldemort" "Fred Weasley"
## [282,] "Lord Voldemort" "George Weasley"
## [283,] "Lord Voldemort" "Ginny Weasley"
## [284,] "Lord Voldemort" "Molly Weasley"
## [285,] "Lord Voldemort" "Ron Weasley"
## [286,] "Severus Snape" "Fred Weasley"
## [287,] "Severus Snape" "George Weasley"
## [288,] "Severus Snape" "Ginny Weasley"
## [289,] "Severus Snape" "Ron Weasley"
## [290,] "Dolores Janes Umbridge" "Ron Weasley"
## [291,] "Arthur Weasley" "Bill Weasley"
## [292,] "Arthur Weasley" "Charlie Weasley"
## [293,] "Arthur Weasley" "Fred Weasley"
## [294,] "Arthur Weasley" "George Weasley"
## [295,] "Arthur Weasley" "Ginny Weasley"
## [296,] "Arthur Weasley" "Molly Weasley"
## [297,] "Arthur Weasley" "Percy Weasley"
## [298,] "Arthur Weasley" "Ron Weasley"
## [299,] "Bill Weasley" "Charlie Weasley"
## [300,] "Bill Weasley" "Fred Weasley"
## [301,] "Bill Weasley" "George Weasley"
## [302,] "Bill Weasley" "Ginny Weasley"
## [303,] "Bill Weasley" "Molly Weasley"
## [304,] "Bill Weasley" "Percy Weasley"
## [305,] "Bill Weasley" "Ron Weasley"
## [306,] "Charlie Weasley" "Fred Weasley"
## [307,] "Charlie Weasley" "George Weasley"
## [308,] "Charlie Weasley" "Ginny Weasley"
## [309,] "Charlie Weasley" "Molly Weasley"
## [310,] "Charlie Weasley" "Percy Weasley"
## [311,] "Charlie Weasley" "Ron Weasley"
## [312,] "Fred Weasley" "George Weasley"
## [313,] "Fred Weasley" "Ginny Weasley"
## [314,] "Fred Weasley" "Molly Weasley"
## [315,] "Fred Weasley" "Percy Weasley"
## [316,] "Fred Weasley" "Ron Weasley"
## [317,] "George Weasley" "Ginny Weasley"
## [318,] "George Weasley" "Molly Weasley"
## [319,] "George Weasley" "Percy Weasley"
## [320,] "George Weasley" "Ron Weasley"
## [321,] "Ginny Weasley" "Molly Weasley"
## [322,] "Ginny Weasley" "Percy Weasley"
## [323,] "Ginny Weasley" "Ron Weasley"
## [324,] "Molly Weasley" "Percy Weasley"
## [325,] "Molly Weasley" "Ron Weasley"
## [326,] "Percy Weasley" "Ron Weasley"
## [327,] "Ron Weasley" "Dobby"
## [328,] "Ron Weasley" "Fluffy"
## [329,] "Ron Weasley" "Aragog"
## [330,] "Ron Weasley" "Grawp"Node importance: Centrality measures
Overall Centrality
Centrality describes the number of edges from nodes. High centrality networks have few nodes with many connections, low centrality networks have many nodes with similar numbers of edges.
For the whole network, we can calculate centrality by degree (centr_degree()), closeness (centr_clo()), eigenvector centrality (centr_eigen()) or betweenness (centr_betw()) of vertices.
In this case, centrality by degree and closeness result in close numbers. However, all centrality measures are below 1, which would mean that the network has many nodes with similar number of edges. This makes sense, as the diameter is only equal to 4 edges, with the average shortest path at 2.
#centrality by degree
centr_degree(g11,mode="total")$centralization## [1] 0.5913462#centrality by closeness
centr_clo(g11, mode="total")$centralization## [1] 0.6047131#centrality by eigenvector
centr_eigen(g11, directed=FALSE)$centralization## [1] 0.717455#centrality by betweenness
centr_betw(g11, "total")$centralization## [1] 0.2697674Plotting Highest Degree Centrality
Visualization of the characters with the highest degree centrality sorted in decreasing order (clockwise). The name of the character is centered in the node (Harry Potter having the highest degree centrality and Nicolas Flamel the lowest).
#layout of RMardown
par(mar=c(1,1,0,0))
visualize_graph(g11 , centrality.type="Degree Centrality")
Network Component Analysis
Centrality is defined as a measure for identifying the most important vertices within a network in graph theory. Several centrality types have been provided to compute central nodes by different formulas, while some analysis are needed to evaluate the most informative ones.
Before further calculating centrality measures, we need to extract the components of the graph to then, identify correctly the recommended the best fitted centrality measures according to this graph.
Segregation of other graph formats
This function extracts the components of other formats of graph. For illustration, we convert the Harry Potter graph to an edge list to be able to use it for this function.
g11_edgelist<-as_edgelist(g11)
misc_extract_components(g11_edgelist)## [[1]]
## IGRAPH 874d4ac DN-- 65 330 --
## + attr: name (v/c)
## + edges from 874d4ac (vertex names):
## [1] Regulus Arcturus Black->Sirius Black
## [2] Regulus Arcturus Black->Vincent Crabbe Sr.
## [3] Regulus Arcturus Black->Bellatrix Lestrange
## [4] Regulus Arcturus Black->Lord Voldemort
## [5] Sirius Black ->Albus Dumbledore
## [6] Sirius Black ->Argus Filch
## [7] Sirius Black ->Hermione Granger
## [8] Sirius Black ->Bellatrix Lestrange
## + ... omitted several edgesCentrality Measure Analysis
All of the introduced centrality measures are not appropriate for all types of networks. So, to figure out which of them is suitable, proper_centralities is specified. This function distinguishes proper centrality types based on network topology. It returns the full names of suitable centrality types for the input graph.
proper_centralities(g11)## [1] "subgraph centrality scores"
## [2] "Topological Coefficient"
## [3] "Average Distance"
## [4] "Barycenter Centrality"
## [5] "BottleNeck Centrality"
## [6] "Centroid value"
## [7] "Closeness Centrality (Freeman)"
## [8] "ClusterRank"
## [9] "Decay Centrality"
## [10] "Degree Centrality"
## [11] "Diffusion Degree"
## [12] "DMNC - Density of Maximum Neighborhood Component"
## [13] "Eccentricity Centrality"
## [14] "Harary Centrality"
## [15] "eigenvector centralities"
## [16] "K-core Decomposition"
## [17] "Geodesic K-Path Centrality"
## [18] "Katz Centrality (Katz Status Index)"
## [19] "Kleinberg's authority centrality scores"
## [20] "Kleinberg's hub centrality scores"
## [21] "clustering coefficient"
## [22] "Lin Centrality"
## [23] "Lobby Index (Centrality)"
## [24] "Markov Centrality"
## [25] "Radiality Centrality"
## [26] "Shortest-Paths Betweenness Centrality"
## [27] "Current-Flow Closeness Centrality"
## [28] "Closeness centrality (Latora)"
## [29] "Communicability Betweenness Centrality"
## [30] "Community Centrality"
## [31] "Cross-Clique Connectivity"
## [32] "Entropy Centrality"
## [33] "EPC - Edge Percolated Component"
## [34] "Laplacian Centrality"
## [35] "Leverage Centrality"
## [36] "MNC - Maximum Neighborhood Component"
## [37] "Hubbell Index"
## [38] "Semi Local Centrality"
## [39] "Closeness Vitality"
## [40] "Residual Closeness Centrality"
## [41] "Stress Centrality"
## [42] "Load Centrality"
## [43] "Flow Betweenness Centrality"
## [44] "Information Centrality"
## [45] "Dangalchev Closeness Centrality"
## [46] "Group Centrality"
## [47] "Harmonic Centrality"
## [48] "Local Bridging Centrality"
## [49] "Wiener Index Centrality"Centrality Computations
In the next step, proper centralities for the network are defined. In order to compute proper centrality types resulting from the proper_centralities, you can use calculate_centralities function as below.
pr_cent<-proper_centralities(g11)## [1] "subgraph centrality scores"
## [2] "Topological Coefficient"
## [3] "Average Distance"
## [4] "Barycenter Centrality"
## [5] "BottleNeck Centrality"
## [6] "Centroid value"
## [7] "Closeness Centrality (Freeman)"
## [8] "ClusterRank"
## [9] "Decay Centrality"
## [10] "Degree Centrality"
## [11] "Diffusion Degree"
## [12] "DMNC - Density of Maximum Neighborhood Component"
## [13] "Eccentricity Centrality"
## [14] "Harary Centrality"
## [15] "eigenvector centralities"
## [16] "K-core Decomposition"
## [17] "Geodesic K-Path Centrality"
## [18] "Katz Centrality (Katz Status Index)"
## [19] "Kleinberg's authority centrality scores"
## [20] "Kleinberg's hub centrality scores"
## [21] "clustering coefficient"
## [22] "Lin Centrality"
## [23] "Lobby Index (Centrality)"
## [24] "Markov Centrality"
## [25] "Radiality Centrality"
## [26] "Shortest-Paths Betweenness Centrality"
## [27] "Current-Flow Closeness Centrality"
## [28] "Closeness centrality (Latora)"
## [29] "Communicability Betweenness Centrality"
## [30] "Community Centrality"
## [31] "Cross-Clique Connectivity"
## [32] "Entropy Centrality"
## [33] "EPC - Edge Percolated Component"
## [34] "Laplacian Centrality"
## [35] "Leverage Centrality"
## [36] "MNC - Maximum Neighborhood Component"
## [37] "Hubbell Index"
## [38] "Semi Local Centrality"
## [39] "Closeness Vitality"
## [40] "Residual Closeness Centrality"
## [41] "Stress Centrality"
## [42] "Load Centrality"
## [43] "Flow Betweenness Centrality"
## [44] "Information Centrality"
## [45] "Dangalchev Closeness Centrality"
## [46] "Group Centrality"
## [47] "Harmonic Centrality"
## [48] "Local Bridging Centrality"
## [49] "Wiener Index Centrality"The following function has the ability to specify some centrality types that are not favored to calculate by the conclude argument. Here, we will select the first ten centrality measures.
calc_cent<-calculate_centralities(g11, include = pr_cent[1:10])Recognition of most informative measures
From this algorithm, the result is a list of computed centralities.
The following is a display of the most informative centrality measures based on principal component analysis. The red line indicates the random threshold of contribution. This barplot represents contribution of variable values based on the number of dimensions.
pca_centralities(calc_cent)
Now that we have the top 10 centrality measures, we will look at the correlation within each centrality measure and choose measures that aren’t highly correlated to each other. When they are highly correlated, it means that they will provide the same kind of information and will only be regarded as repetitive.
Correlation between computed centrality measures
To comprehend pair correlation among centralities, visualize_correlations method is appropriate; it is able to specify the type of correlation which are interesting to obtain and analyse.
visualize_correlations(calc_cent,"pearson")
#A display of correlation among computed centrality measures. The red to blue highlighted circles represent the top to bottom Pearson correlation coefficients(Benesty et al. 2009) which differ from -1 to 1. The higher the value becomes larger, circles' sizes get larger too.As we can see, there are high levels of correlations between:
- Closeness, Decay and Barycenter; these were the top 3 measures from which we will only choose closeness
- Degree and subgraph; choose degree
- Topological
- Bottleneck
Chosen Centrality Measures:
Closeness
Closeness centrality measures the average distance from any given vertex to all other vertices. The closeness list is dominated exclusively by well known characters, most of them being aligned as good and from Gryffindor. This is an indicator that House and Alignment are closely aligned (connected).
#create a new dataframe for top 20 centralities from the keys dataframe
keys_top20_centrality<-keys
#keep only important columns to show
keys_top20_centrality$Color<-NULL
keys_top20_centrality$id<-NULL
keys_top20_centrality$Gender_Male<-NULL
#assign closeness value to each actor
keys_top20_centrality$closeness <- igraph::closeness(g11, mode = "total")
#order the dataframe based on decreasing order in closeness
head(keys_top20_centrality[order(-keys_top20_centrality$closeness),],n=20)## Character Alignment House closeness
## 40 Harry Potter Good Gryffindor 0.012500000
## 59 Ron Weasley Good Gryffindor 0.010989011
## 22 Hermione Granger Good Gryffindor 0.010869565
## 46 Lord Voldemort Evil Slytherin 0.010752688
## 12 Albus Dumbledore Good Gryffindor 0.010204082
## 2 Sirius Black Good Gryffindor 0.009259259
## 56 Ginny Weasley Good Gryffindor 0.008928571
## 54 Fred Weasley Good Gryffindor 0.008849558
## 55 George Weasley Good Gryffindor 0.008849558
## 57 Molly Weasley Good Gryffindor 0.008849558
## 29 Neville Longbottom Good Gryffindor 0.008695652
## 39 Peter Pettigrew Evil Gryffindor 0.008695652
## 23 Rubeus Hagrid Good Gryffindor 0.008620690
## 48 Severus Snape Good Slytherin 0.008620690
## 26 Bellatrix Lestrange Evil Slytherin 0.008547009
## 52 Bill Weasley Good Gryffindor 0.008547009
## 53 Charlie Weasley Good Gryffindor 0.008474576
## 32 Remus Lupin Good Gryffindor 0.008403361
## 37 Minerva McGonagall Good Gryffindor 0.008403361
## 34 Lucius Malfoy Evil Slytherin 0.008333333Highest Degree
The top 20 list with the highest degree centrality shows the actors with the most links in relation to nodes within the network. In this case, a higher centrality degree is directly related to the number of relationships the characters have with others. Although Hermione is the character with the most relationships (most degrees) and not Harry Potter, because she’s considered the most knowledgeable character in the story line.
Again, the high degree centrality of the top-20 list might be explained by virtue of the dominance of the common House. Furthermore, the dominance of Good Alignment in characters suggests that there are fewer Evil characters than Good characters. If it is assumed that Evil characters are nevertheless required in all Harry Potter films, but considering the movies are on average PG-13, Good characters are more popular. This could be a possible explanation for the high degree values of the top 20 list.
#assign degree value to each actor
keys_top20_centrality$degree<-igraph::degree(g1,mode = "total")
#order the dataframe based on decreasing order in degree
head(keys_top20_centrality[order(-keys_top20_centrality$degree),],n=20)## Character Alignment House closeness degree
## 22 Hermione Granger Good Gryffindor 0.010869565 59
## 59 Ron Weasley Good Gryffindor 0.010989011 57
## 40 Harry Potter Good Gryffindor 0.012500000 56
## 12 Albus Dumbledore Good Gryffindor 0.010204082 55
## 46 Lord Voldemort Evil Slytherin 0.010752688 46
## 55 George Weasley Good Gryffindor 0.008849558 35
## 51 Arthur Weasley Good Gryffindor 0.008333333 34
## 2 Sirius Black Good Gryffindor 0.009259259 33
## 56 Ginny Weasley Good Gryffindor 0.008928571 30
## 52 Bill Weasley Good Gryffindor 0.008547009 28
## 54 Fred Weasley Good Gryffindor 0.008849558 28
## 53 Charlie Weasley Good Gryffindor 0.008474576 27
## 57 Molly Weasley Good Gryffindor 0.008849558 24
## 37 Minerva McGonagall Good Gryffindor 0.008403361 23
## 39 Peter Pettigrew Evil Gryffindor 0.008695652 23
## 48 Severus Snape Good Slytherin 0.008620690 23
## 58 Percy Weasley Good Gryffindor 0.007633588 23
## 23 Rubeus Hagrid Good Gryffindor 0.008620690 21
## 38 Alastor Moody Good <NA> 0.008333333 19
## 41 James Potter Good Gryffindor 0.008064516 18Topological Coefficient
The topological coefficient is a relative measure for the extent to which a node shares neighbors with other nodes. More precisely, the topological coefficient is the number of neighbors shared between a pair of nodes, x and y, plus one if there is a direct link between them, divided by the number of neighbors of node x. Thus, nodes that have one or no neighbors are assigned a topological coefficient of 0.
In this case, it is normal for Harry Potter, Voldemort, Hermione and Ron to have the highest values, as they share many of the same relationships with other characters and most of the scenes include one of the characters.
#assign topological coefficient value to each actor
keys_top20_centrality$topological<-1/centiserve::topocoefficient(g11)
#order the dataframe based on decreasing order of coefficient
head(keys_top20_centrality[order(-keys_top20_centrality$topological),],n=20)## Character Alignment House closeness degree
## 40 Harry Potter Good Gryffindor 0.012500000 56
## 46 Lord Voldemort Evil Slytherin 0.010752688 46
## 22 Hermione Granger Good Gryffindor 0.010869565 59
## 59 Ron Weasley Good Gryffindor 0.010989011 57
## 12 Albus Dumbledore Good Gryffindor 0.010204082 55
## 29 Neville Longbottom Good Gryffindor 0.008695652 17
## 5 Vincent Crabbe Sr. Evil Slytherin 0.007246377 11
## 2 Sirius Black Good Gryffindor 0.009259259 33
## 7 Bartemius Crouch Sr. Good <NA> 0.006250000 6
## 23 Rubeus Hagrid Good Gryffindor 0.008620690 21
## 26 Bellatrix Lestrange Evil Slytherin 0.008547009 16
## 24 Igor Karkaroff Neutral Durmstrang 0.006756757 5
## 34 Lucius Malfoy Evil Slytherin 0.008333333 15
## 56 Ginny Weasley Good Gryffindor 0.008928571 30
## 9 Fleur Delacour Good Beauxbatons 0.008130081 12
## 57 Molly Weasley Good Gryffindor 0.008849558 24
## 8 Bartemius Crouch Jr. Evil <NA> 0.008196721 12
## 58 Percy Weasley Good Gryffindor 0.007633588 23
## 39 Peter Pettigrew Evil Gryffindor 0.008695652 23
## 33 Draco Malfoy Evil Slytherin 0.007812500 14
## topological
## 40 5.440585
## 46 4.779443
## 22 4.750515
## 59 4.607004
## 12 4.280702
## 29 3.834783
## 5 3.735849
## 2 3.490050
## 7 3.448276
## 23 3.325670
## 26 3.289796
## 24 3.288889
## 34 3.200000
## 56 2.983957
## 9 2.929907
## 57 2.919668
## 8 2.887701
## 58 2.882353
## 39 2.879257
## 33 2.861272BottleNeck Centrality
A bottleneck to a network is a node that has information to the network that is concentrated. The bottleneckness of a node can be calculated using betweenness centrality, which is a measure of a node’s centrality in a network, and equal to the number of shortest paths going through it.
In other words, a node characterized by high bottleneck centrality means that most nodes need to pass through this specific node to follow a certain path (these paths being the shortest paths). Thus, even though Hermione might have the most connections, Harry has more particular connections which requires the information to pass through him instead of Hermione.
#assign degree value to each bottleneck centrality value
keys_top20_centrality$bottleneck <- bottleneck(g11)
#order the dataframe based on decreasing order of bottleneck centrality value
head(keys_top20_centrality[order(-keys_top20_centrality$bottleneck),],n=20)## Character Alignment House closeness degree
## 40 Harry Potter Good Gryffindor 0.012500000 56
## 46 Lord Voldemort Evil Slytherin 0.010752688 46
## 22 Hermione Granger Good Gryffindor 0.010869565 59
## 59 Ron Weasley Good Gryffindor 0.010989011 57
## 12 Albus Dumbledore Good Gryffindor 0.010204082 55
## 2 Sirius Black Good Gryffindor 0.009259259 33
## 5 Vincent Crabbe Sr. Evil Slytherin 0.007246377 11
## 29 Neville Longbottom Good Gryffindor 0.008695652 17
## 26 Bellatrix Lestrange Evil Slytherin 0.008547009 16
## 30 Luna Lovegood Good Ravenclaw 0.007518797 9
## 58 Percy Weasley Good Gryffindor 0.007633588 23
## 23 Rubeus Hagrid Good Gryffindor 0.008620690 21
## 34 Lucius Malfoy Evil Slytherin 0.008333333 15
## 35 Narcissa Malfoy Evil Slytherin 0.006896552 7
## 17 Seamus Finnigan Good Gryffindor 0.007633588 10
## 24 Igor Karkaroff Neutral Durmstrang 0.006756757 5
## 33 Draco Malfoy Evil Slytherin 0.007812500 14
## 49 Nymphadora Tonks Good Hufflepuff 0.006711409 6
## 56 Ginny Weasley Good Gryffindor 0.008928571 30
## 1 Regulus Arcturus Black Good Slytherin 0.006802721 5
## topological bottleneck
## 40 5.440585 62
## 46 4.779443 59
## 22 4.750515 49
## 59 4.607004 48
## 12 4.280702 34
## 2 3.490050 11
## 5 3.735849 9
## 29 3.834783 8
## 26 3.289796 6
## 30 2.181818 6
## 58 2.882353 6
## 23 3.325670 5
## 34 3.200000 4
## 35 2.311828 4
## 17 2.494118 3
## 24 3.288889 3
## 33 2.861272 3
## 49 1.904762 3
## 56 2.983957 3
## 1 2.074074 2Other Common Centrality Measures
During our Social Network Analysis Class, we had the opportunity to learn about other common centrality measures that weren’t considered important through the PCA analysis. However, we did want to analyze these measures, as they were essential to our learning curve during our semester.
Betweenness
Betweenness centrality is a metric used to find nodes that serve as a bridge (connector) from one part of the network to another. In other words, betweenness measures the average length of shortest paths that pass through a vertex. Accordingly characters with a high betweenness can be understood as links between different closed clusters of characters.
As we can see, only a few characters have high level of betweenness. On average, if characters have low betweenness measurements, this demonstrates that most of these nodes do not lie on the shortest paths. This goes in hand with the bottleneck centrality measure, where characters that lie more frequently on these paths have a higher probability of being considered bottlenecks for the network (which is true when looking at the first two values of each dataframe).
#create a new dataframe for top 20 centralities from the keys dataframe
keys_top20_others <- keys
#keep only important columns to show
keys_top20_others$Color<-NULL
keys_top20_others$id<-NULL
keys_top20_others$Gender_Male<-NULL
#assign degree value to each bottleneck centrality value
keys_top20_others$betweenness <- igraph::betweenness(g11, directed = FALSE)
#order the dataframe based on decreasing order of bottleneck centrality value
head(keys_top20_others[order(-keys_top20_others$betweenness),],n=20)## Character Alignment House betweenness
## 40 Harry Potter Good Gryffindor 568.39191
## 46 Lord Voldemort Evil Slytherin 410.15172
## 12 Albus Dumbledore Good Gryffindor 202.59718
## 59 Ron Weasley Good Gryffindor 198.58285
## 22 Hermione Granger Good Gryffindor 189.48922
## 29 Neville Longbottom Good Gryffindor 90.23966
## 30 Luna Lovegood Good Ravenclaw 63.00000
## 2 Sirius Black Good Gryffindor 56.19830
## 23 Rubeus Hagrid Good Gryffindor 38.09892
## 26 Bellatrix Lestrange Evil Slytherin 34.39321
## 8 Bartemius Crouch Jr. Evil <NA> 26.77249
## 5 Vincent Crabbe Sr. Evil Slytherin 20.69743
## 58 Percy Weasley Good Gryffindor 20.28649
## 34 Lucius Malfoy Evil Slytherin 19.94463
## 9 Fleur Delacour Good Beauxbatons 16.34125
## 39 Peter Pettigrew Evil Gryffindor 13.27643
## 56 Ginny Weasley Good Gryffindor 13.02037
## 21 Gregory Goyle Evil Slytherin 12.09342
## 48 Severus Snape Good Slytherin 12.09215
## 24 Igor Karkaroff Neutral Durmstrang 10.14327When plotting the higher values of betweenness in our inital network:
#layout of RMarkdown
par(mar=c(1,1,1,1))
#create variables for the plot
edge_betweenness <- igraph::edge_betweenness(g11, directed = FALSE)
betweenness<-igraph::betweenness(g11, directed = FALSE)
plot(igraph::simplify(g1,edge.attr.comb = "min"),
layout = mylayout,
legend = TRUE,
vertex.label = V(g1)$Character,
vertex.color = V(g1)$Color,
vertex.size = betweenness * 0.001,
vertex.frame.color = "gray",
vertex.label.color = "black",
vertex.label.cex = 0.6,
edge.width = edge_betweenness * 0.12,
edge.color = "gray",
edge.lty = E(g1)$lty,
asp = 0.75,
rescale = FALSE,
ylim = c(-4.5,4.5),
xlim = c(-2,2))
legend("top", legend = "", cex = 0.95, bty = "n", ncol = 1,
title = "Harry Potter's Character Edge Betweenness")
Top 10 Strongest 2-Character Betweenness
Having a closer look at the betweenness topic, we can also explore the strongest 2-character betweenness. The 2-Character betweenness represents the top 10 most popular relationships where the most frequent shortest paths pass through. From this dataframe, we can infer that the highest values are given to characters that have strong relationships and one on the characters in the relationship, is only connected to the other character (who then has a high degree centrality). For example, Harry Potter and Hedwig have a high in betweenness because Hedwig is only connected to Harry, but Harry is one of the most connected characters.
data.frame(edge = attr(E(g11), "vnames"),
betweenness = edge_betweenness) %>%
tibble::rownames_to_column() %>%
arrange(-betweenness) %>%
.[1:10, ]## rowname edge betweenness
## 1 85 Albus Dumbledore|Nicolas Flamel 64.00000
## 2 205 Luna Lovegood|Xenophilius Lovegood 64.00000
## 3 265 Harry Potter|Hedwig 64.00000
## 4 266 Harry Potter|Moaning Myrtle 64.00000
## 5 275 Tom Riddle Sr.|Lord Voldemort 63.00000
## 6 276 Mary Riddle|Lord Voldemort 63.00000
## 7 197 Alice Longbottom|Neville Longbottom 54.47857
## 8 199 Frank Longbottom|Lord Voldemort 54.03512
## 9 186 Igor Karkaroff|Lord Voldemort 53.75210
## 10 206 Luna Lovegood|Harry Potter 51.28337Graph Theory
Eigenvector
Eigenvector centrality is a measure of the influence of a node in a network. Relative scores are assigned to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. A high eigenvector score means that a node is connected to many nodes who themselves have high scores. PageRank and the Katz centrality are variants of the eigenvector centrality. PageRank won’t be computed as it is for often used to rank web pages and for directed graphs.
Eigenvector centrality works well only if the graph is (strongly) connected. Real undirected networks typically have a large connected component, of size proportional to the network size. Therefore, because our graph is undirected, we do not need to find the Katz centrality measure, as it is used mostly for directed graphs.
If a directed network is not strongly connected, only vertices that are in strongly connected components or in the out-component of such components can have non-zero eigenvector centrality. The other vertices, such as those in the in-components of strongly connected components, all have, with little justification, null centrality. This happens because nodes with no incoming edges have, by definition, a null eigenvector centrality score, and so have nodes that are pointed to by only nodes with a null centrality score.
Katz centrality: A way to work around this problem is to give each node a small amount of centrality for free, regardless of the position of the vertex in the network. Hence, each node has a minimum, positive amount of centrality that it can transfer to other nodes by referring to them.
However, because we have an undirected network, we only computed the eigenvector centralities. From our results, we can see that the top characters are from the Weasley family. This family has ties with important characters and are considered one of the prominent wizarding families.
#assign degree value to each bottleneck centrality value
keys_top20_others$eigen<-igraph::eigen_centrality(g1)$vector
#order the dataframe based on decreasing order of bottleneck centrality value
head(keys_top20_others[order(-keys_top20_others$eigen),],n=20)## Character Alignment House betweenness eigen
## 22 Hermione Granger Good Gryffindor 189.489217 1.0000000
## 55 George Weasley Good Gryffindor 6.508347 0.9511415
## 59 Ron Weasley Good Gryffindor 198.582847 0.9288641
## 12 Albus Dumbledore Good Gryffindor 202.597178 0.9242241
## 51 Arthur Weasley Good Gryffindor 6.616091 0.9238878
## 52 Bill Weasley Good Gryffindor 3.465663 0.8077169
## 53 Charlie Weasley Good Gryffindor 1.807752 0.7994759
## 56 Ginny Weasley Good Gryffindor 13.020372 0.7737188
## 40 Harry Potter Good Gryffindor 568.391905 0.7595342
## 54 Fred Weasley Good Gryffindor 6.508347 0.7492480
## 46 Lord Voldemort Evil Slytherin 410.151718 0.6487222
## 57 Molly Weasley Good Gryffindor 8.878000 0.6291314
## 58 Percy Weasley Good Gryffindor 20.286493 0.6268225
## 2 Sirius Black Good Gryffindor 56.198299 0.6151905
## 37 Minerva McGonagall Good Gryffindor 4.955393 0.5625002
## 48 Severus Snape Good Slytherin 12.092155 0.5270981
## 38 Alastor Moody Good <NA> 2.959805 0.4834333
## 39 Peter Pettigrew Evil Gryffindor 13.276428 0.4563157
## 50 Dolores Janes Umbridge Evil Slytherin 2.850000 0.4167115
## 43 Quirinus Quirrell Evil Ravenclaw 0.000000 0.4081002Network Measures: Transitivity and Reciprocity
There is no reciprocity for undirected graph. Thus, we will only compute transitivity.
Transitivity
Transitivity measures the probability that the adjacent vertices of a vertex are connected. This is sometimes also called the clustering coefficient. The clustering coefficient measures the transitivity of a network, that is how connected closed groups of vertices are with each other. The global clustering coefficient gives an indication of the overall clustering in the network, while the local clustering coefficient gives an indication of the embeddedness of single nodes.
This suggests that there are more tightly knit groups of actors by virtue of shared Alignment.
#calculate global transitivity in the dataframe
transitivity(g11, type="global")## [1] 0.4133759#calculate local transitivity in the dataframe
transitivity(g11, type="localaverageundirected")## [1] 0.6518034Network Models: Real World networks
Erdos-Renyi (random graphs)
In this section, we will explore the Erdos-Renyi model and compare its structural properties to those of real-world networks that we graphed above.
#For our Erdos-Renyi model, we used the sample_gnm function from which we used the number of nodes in the networks to create our new models.
vertices_characters<-vcount(g11)
edges_characters<-ecount(g11)
er_potter<-sample_gnm(vertices_characters,edges_characters, directed = FALSE)Degree distribution of Erdos-Renyi (ER) is now normally distributed versus the real-world networks (original distribution of Harry Potter’s network was right skewed), where degrees between 5 and 15 are the most frequent.
#Degree Distribution
deg_erpotter<-igraph::degree(er_potter)
hist(igraph::degree(er_potter), xlim = c(0,20), ylim = c(0,20), col = "tomato")
The average path length and diameter have not changed between both model, with 2 and 4. These values may not change when comparing smaller networks together; both networks are as well connected.
#Network Diameter and Average Path Length
mean_distance(er_potter, directed=F)## [1] 2.020192diameter(er_potter)## [1] 4When applying the Erdos-Renyi models, the local and global clustering coefficient becomes almost equal to 0, removing any types of triangular connections or relationships that existed between the nodes whether local or global. Furthermore, the almost 0 values for the cluster coefficient for the Erdos-Rényi model demonstrates a lack of clusters in this network. This signifies that clusters found in the real world networks would not necessarily occur given random conditions.
#(Global and Local) Clustering Coefficient
global_er_potter<-transitivity(er_potter) #GLOBAL
global_er_potter## [1] 0.1563253local_er_potter<-transitivity(er_potter, type='localaverageundirected') #LOCAL
local_er_potter## [1] 0.1635723In conclusion, the Erdos-Renyi model doesn’t describe real life networks as well. Access to larger networks (social networks, power networks, the world wide web, etc.) have made clear that many previous graph-theory models which were used to approximate networks were incorrect - the most famous of which, the Erdos-Renyi model, failed to replicate the clustering, triadic closure, and hubs seen in real-world networks. As a result, there were two notable models created in an attempt to fix some of the problems Erdos-Renyi had: The Watts-Strogatz model and the Barabasi-Albert model.
Watts Strogatz (small world)
The Watts-Strogatz model is a random graph generation model that produces graphs with small-world properties, including short average path lengths and high clustering.
They start with the lattice model that has high clustering and high mean path length. Then, add to the model a probability p that an edge is rewired, meaning that the edge is disconnected from one of its nodes and then randomly connected to another node anywhere in the network. Each edge is chosen to be rewired independent with probability p.
#create small world network and plot to see what it looks like
g.ws <- sample_smallworld(dim=2, size=65, nei=1, p=0.015)
plot(g.ws, vertex.size=6, vertex.label=NA, layout=layout_in_circle)
When the probability p is low, then most connections are still the original local connections that they connect nodes that are nearby in the lattice. However, some of the edges that have been rewired might turn into long distance connections that connect nodes that are far away from each other in the lattice. These long distance connections, or shortcuts, immediately create a short distance between the nodes around either end of the shortcut.
Originally, each node was symmetrically connected to its nearest neighbors along the ring. But then, each edge was rewired with probability 1/65. If an edge was selected for rewiring, one end of the edge was disconnected from a node and reconnected with a randomly chosen node. When probability is large (like in our graph), the transivity becomes low.
This model does not replicate the real world network, as it’s mean path length is around 5 times longer. This is because of the rewiring of the edges, which creates longer paths on average.
transitivity(g.ws) #clustering coefficient## [1] 0.0002345307mean_distance(g.ws) #diameter## [1] 12.0706Preferential Attachment: Barabasi
The Barabási-Albert (BA) model is an algorithm for generating random scale-free networks using a preferential attachment mechanism. The BA model tries to explain the existence of few nodes with high degree in real networks.
par(mar=c(1,1,1,1))
#create BA network and plot to see what it looks like
g.ba <- sample_pa(n=65, power=1, m=1, directed=F)
plot(g.ba, vertex.size=6, vertex.label=NA)
Many observed networks (at least approximately) fall into the class of scale-free networks, meaning that they have power-law (or scale-free) degree distributions, while random graph models such as the Erdos-Rényi (ER) model and the Watts-Strogatz (WS) model do not exhibit power laws.
Even though the BA’s network degree distribution has the same right skewed shape, there are too many node degrees between 0 and 2 degrees. In addition, the BA model doesn’t represent the outliers that we have in the real world networks (nodes of degrees of 50 for example).
hist(igraph::degree(g.ba), col="tomato", xlab="Degree", ylab="Frequency", main="", ylim=c(0,60), xlim=c(1,10))
Information Diffusion
Group Centrality Measure
The basic idea of measuring the group-level centrality is to treat a group of nodes as a large pseudo-node. We propose several methods to measure the tie status between this pseudo node and other nodes, responding to several common edge value interpretations (An and Liu, 2015).
In a context, centrality measures may be insufficient to describe the data.
None of these packages (network, igraph, statnet) provide a comprehensive toolbox to calculate group centrality measures and to identify key players, who constitute the most central group, in a network. Determining the key players in a network is very important because interventions rely on key players to facilitate the intervention.
The algorithm for identifying key players in package keyplayer essentially consists of three steps. First, users choose a metric to measure centrality in a network. Second, the algorithm (specifically the kpcent function) will randomly pick a group of players and measure their group centrality. Third, the algorithm (specifically the kpset function) will select the group of players with the highest group centrality as the desired key players.
The keys players, 15 and 32, in our network belong to Vernon Dursley and Remus Lupin. This means that if a message needed to be passed around within the network, the speed at which all characters would know the message would be the highest with these characters. This is surprising, as both these characters are not the main characters of the story.
kpset(g11_ajd, size = 2, type = "diffusion", cmode = "all", method = "max")$keyplayers## [1] 15 32kpset(g11_ajd, size = 2, type = "diffusion", cmode = "all", M = 1, binary = TRUE)$keyplayers## [1] 15 32When calculating the group-level centraility scores of both keyplayer nodes 15 and 32, we quickly realize that node 15 has a much higher score than 32. In addition, other nodes that aren’t considered keyplayers also have higher scores.
#kpcent reports the group-level centrality scores.
kpcent(g11_ajd, 15, type = "degree", binary = TRUE) #for 15## [1] 96kpcent(g11_ajd, 32, type = "degree", binary = TRUE) #for 32## [1] 8kpcent(g11_ajd,c(15,32),type="degree") #for both together## [1] 102Thus, we want to confirm these keyplayers through another diffusion function.
Diffusion
Now, we wanted to compare the diffusion level of the same network with another function. Diffusion measures player’s ability to disseminate information through all the possible paths. For each path from i to j there is a reaching probability P(ij), which is specified in the inputted adjacency matrix (g11_ajd).
From both algorithms, we can see that we obtain two similar results, as both functions determine Vernon Dursley as the character with the highest diffusion. However, Remus Lupin is not considered a key player for diffusion in the following function.
[We are going to test these results in Netlogo]
keys_diffusion<-keys
keys_diffusion$Gender_Male<-NULL
keys_diffusion$Color<-NULL
keys_diffusion$id<-NULL
keys_diffusion$diffusion<-diffusion(g11_ajd, c(1:65), T = ncol(g11_ajd))
head(keys_diffusion[order(-keys_diffusion$diffusion),], n=65 )## Character Alignment House diffusion
## 15 Vernon Dursley Neutral Muggle 4.977681e+80
## 25 Viktor Krum Good Durmstrang 4.542946e+80
## 8 Bartemius Crouch Jr. Evil <NA> 4.310119e+80
## 6 Vincent Crabbe Evil Slytherin 4.030147e+80
## 5 Vincent Crabbe Sr. Evil Slytherin 4.009859e+80
## 2 Sirius Black Good Gryffindor 3.382735e+80
## 23 Rubeus Hagrid Good Gryffindor 3.107142e+80
## 21 Gregory Goyle Evil Slytherin 3.071350e+80
## 22 Hermione Granger Good Gryffindor 3.071350e+80
## 24 Igor Karkaroff Neutral Durmstrang 2.946502e+80
## 20 Goyle Sr. Evil Slytherin 2.587286e+80
## 14 Petunia Dursley Neutral Muggle 2.552084e+80
## 43 Quirinus Quirrell Evil Ravenclaw 2.461041e+80
## 18 Nicolas Flamel Good Beauxbatons 2.459118e+80
## 53 Charlie Weasley Good Gryffindor 2.368910e+80
## 7 Bartemius Crouch Sr. Good <NA> 2.275109e+80
## 9 Fleur Delacour Good Beauxbatons 2.137745e+80
## 12 Albus Dumbledore Good Gryffindor 1.998798e+80
## 50 Dolores Janes Umbridge Evil Slytherin 1.970880e+80
## 13 Dudley Dursley Neutral Muggle 1.965585e+80
## 4 Cho Chang Good Ravenclaw 1.810983e+80
## 39 Peter Pettigrew Evil Gryffindor 1.705356e+80
## 10 Cedric Diggory Good Hufflepuff 1.691566e+80
## 40 Harry Potter Good Gryffindor 1.679720e+80
## 28 Frank Longbottom Good Gryffindor 1.667300e+80
## 16 Argus Filch Neutral Squib 1.435114e+80
## 37 Minerva McGonagall Good Gryffindor 1.403734e+80
## 36 Olympe Maxime Good Beauxbatons 1.402095e+80
## 17 Seamus Finnigan Good Gryffindor 1.364358e+80
## 27 Alice Longbottom Good Gryffindor 1.247948e+80
## 29 Neville Longbottom Good Gryffindor 1.241279e+80
## 35 Narcissa Malfoy Evil Slytherin 1.236057e+80
## 34 Lucius Malfoy Evil Slytherin 1.189814e+80
## 45 Mary Riddle Good Muggle 1.141611e+80
## 44 Tom Riddle Sr. Evil Muggle 1.102753e+80
## 31 Xenophilius Lovegood Good Ravenclaw 1.074551e+80
## 51 Arthur Weasley Good Gryffindor 1.005567e+80
## 55 George Weasley Good Gryffindor 9.750868e+79
## 49 Nymphadora Tonks Good Hufflepuff 9.513838e+79
## 56 Ginny Weasley Good Gryffindor 9.169994e+79
## 57 Molly Weasley Good Gryffindor 9.169994e+79
## 58 Percy Weasley Good Gryffindor 9.169994e+79
## 3 Lavender Brown Good Gryffindor 8.578294e+79
## 11 Aberforth Dumbledore Good Gryffindor 7.039974e+79
## 41 James Potter Good Gryffindor 6.914750e+79
## 26 Bellatrix Lestrange Evil Slytherin 6.829383e+79
## 46 Lord Voldemort Evil Slytherin 6.637112e+79
## 19 Cornelius Fudge Neutral <NA> 6.581592e+79
## 30 Luna Lovegood Good Ravenclaw 6.337262e+79
## 47 Rita Skeeter Good <NA> 5.888776e+79
## 1 Regulus Arcturus Black Good Slytherin 5.838832e+79
## 52 Bill Weasley Good Gryffindor 5.227420e+79
## 54 Fred Weasley Good Gryffindor 4.759289e+79
## 42 Lily Potter Good Gryffindor 4.639607e+79
## 33 Draco Malfoy Evil Slytherin 3.515288e+79
## 60 Dobby Good <NA> 3.362040e+79
## 38 Alastor Moody Good <NA> 3.141697e+79
## 62 Hedwig Good <NA> 2.888647e+79
## 63 Moaning Myrtle Good Ravenclaw 2.888647e+79
## 32 Remus Lupin Good Gryffindor 2.484401e+79
## 64 Aragog Neutral <NA> 2.470360e+79
## 65 Grawp Good <NA> 2.470360e+79
## 48 Severus Snape Good Slytherin 2.338774e+79
## 59 Ron Weasley Good Gryffindor 1.162673e+79
## 61 Fluffy Good <NA> 5.658621e+78Community Detection
Subgroups and commmunities
Subgroup discovery aims at identifying interesting descriptive subgroups contained in a dataset - from a compositional network analysis view, aiming at a description given by a set of attribute values.
The following functions for cliques give us:
- The list of all cliques (groups of nodes) within the network
- The size of all the cliques
- The largest or all the maximal cliques in an undirected graph
In this case, we have two maximal cliques, which differ very slightly (Arthur Weasley and Voldemort are interchanged). This could be because they are within the limits of being included in the clique. Thus, taking them out of the cliques would not have such a great impact.
sapply(cliques(g11), length) #clique sizes## [1] 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 2 1 2 1 2 1 2
## [24] 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 2
## [47] 3 2 3 2 3 3 2 2 1 2 3 2 2 1 2 3 2 3 2 2 2 1 2
## [70] 2 1 2 3 2 3 2 1 2 3 2 1 2 3 4 3 4 3 2 3 4 3 2
## [93] 3 4 3 4 3 2 3 2 3 2 3 2 3 3 2 3 2 3 3 2 2 2 2
## [116] 1 2 3 4 3 4 4 3 3 2 3 2 3 2 3 3 2 2 2 1 2 3 4
## [139] 3 2 3 4 3 2 3 2 1 2 3 4 3 2 2 3 2 1 2 3 4 3 2
## [162] 3 2 1 2 3 4 3 2 3 2 1 2 3 4 3 2 3 2 1 2 3 4 3
## [185] 2 3 2 1 2 2 2 2 1 2 3 4 5 4 3 3 4 3 2 3 4 5 4
## [208] 5 4 3 4 3 4 3 3 2 3 4 5 4 3 4 5 4 5 4 3 4 3 4
## [231] 3 4 3 2 3 2 2 3 4 3 3 3 2 3 4 3 3 2 3 4 3 4 3
## [254] 4 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 2 2 2 2 2 1
## [277] 2 3 4 5 4 3 4 3 2 3 4 3 2 3 2 1 2 3 4 5 4 3 4
## [300] 4 3 4 4 3 3 2 3 4 3 4 3 2 3 4 3 2 3 3 2 3 3 2
## [323] 2 1 2 3 4 5 4 3 4 3 2 3 4 3 4 3 2 3 2 3 2 2 1
## [346] 2 3 4 5 5 4 4 3 4 4 3 3 2 3 4 4 3 3 2 3 3 2 2
## [369] 1 2 3 4 5 4 3 4 3 2 3 4 3 2 3 2 2 1 2 3 4 5 4
## [392] 3 4 3 2 3 4 5 4 3 4 3 2 3 4 3 2 3 2 1 2 3 4 5
## [415] 6 5 4 5 4 3 4 5 4 3 4 5 4 3 4 3 2 3 4 5 6 5 4
## [438] 5 4 3 4 5 4 3 4 3 2 3 4 5 6 5 4 5 6 5 4 5 4 5
## [461] 4 3 4 5 4 4 3 4 5 4 4 3 4 5 4 3 3 4 3 4 3 3 3
## [484] 2 3 4 5 4 3 4 5 4 3 4 3 4 3 2 3 4 3 3 2 3 4 3
## [507] 3 2 3 4 3 2 2 3 2 3 2 2 2 1 2 3 4 5 6 5 4 5 5
## [530] 4 4 3 4 5 4 3 4 4 3 3 2 3 4 5 6 5 4 5 4 3 4 5
## [553] 4 3 4 3 2 3 4 5 4 3 4 4 3 3 2 3 4 3 2 3 3 2 2
## [576] 1 2 3 4 5 6 5 4 5 4 3 4 5 4 3 4 3 2 3 4 5 6 5
## [599] 4 5 4 3 4 5 6 5 4 5 4 3 4 5 4 3 4 3 2 3 2 3 4
## [622] 5 4 3 4 3 2 3 4 5 4 3 4 3 2 3 4 5 4 3 4 3 2 3
## [645] 4 5 4 3 4 3 2 3 4 5 4 3 4 3 3 2 3 4 3 2 3 2 2
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## [6855] 8 8 7 8 9 8 7 8 8 7 7 6 7 8 9 10 9 8 9 8 7 8 9
## [6878] 8 7 8 7 6 7 8 9 8 7 8 8 7 7 6 7 8 7 6 7 7 6 6
## [6901] 5 6 7 8 9 8 7 8 8 7 7 6 7 8 7 6 7 7 6 6 5 6 7
## [6924] 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 7 7 6 6
## [6947] 5 6 7 6 5 6 6 5 5 4 5 6 7 8 9 8 7 8 8 7 7 6 7
## [6970] 8 7 6 7 7 6 6 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6
## [6993] 5 6 7 8 7 6 7 7 6 6 5 6 7 6 5 6 6 5 5 4 5 6 7
## [7016] 8 7 6 7 7 6 6 5 6 7 6 5 6 6 5 5 4 5 6 7 8 7 6
## [7039] 7 6 5 6 7 6 5 6 5 4 5 6 7 6 5 6 6 5 5 4 5 6 5
## [7062] 4 5 5 4 4 3 4 5 6 7 8 9 8 7 8 8 7 7 6 7 8 7 6
## [7085] 7 7 6 6 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7
## [7108] 8 7 6 7 7 6 6 5 6 7 6 5 6 6 5 5 4 5 6 7 8 7 6
## [7131] 7 7 6 6 5 6 7 6 5 6 6 5 5 4 5 6 7 8 7 6 7 6 5
## [7154] 6 7 6 5 6 5 4 5 6 7 6 5 6 6 5 5 4 5 6 5 4 5 5
## [7177] 4 4 3 4 5 6 7 8 7 6 7 7 6 6 5 6 7 6 5 6 6 5 5
## [7200] 4 5 6 7 8 7 6 7 6 5 6 7 6 5 6 5 4 5 6 7 6 5 6
## [7223] 6 5 5 4 5 6 5 4 5 5 4 4 3 4 5 6 7 6 5 6 6 5 5
## [7246] 4 5 6 5 4 5 5 4 4 3 4 5 6 7 6 5 6 5 4 5 6 5 4
## [7269] 5 4 3 4 5 6 5 4 5 5 4 4 3 4 5 4 3 4 4 3 3 2 3
## [7292] 4 5 6 7 8 9 8 7 8 8 7 7 6 7 8 7 6 7 7 6 6 5 6
## [7315] 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 7 7 6
## [7338] 6 5 6 7 6 5 6 6 5 5 4 5 6 7 6 5 6 5 4 5 6 7 8
## [7361] 7 6 7 7 6 6 5 6 7 6 5 6 6 5 5 4 5 6 7 8 7 6 7
## [7384] 6 5 6 7 6 5 6 5 4 5 6 7 6 5 6 6 5 5 4 5 6 5 4
## [7407] 5 5 4 4 3 4 5 6 7 8 7 6 7 7 6 6 5 6 7 6 5 6 6
## [7430] 5 5 4 5 6 7 8 7 6 7 6 5 6 7 6 5 6 5 4 5 6 7 6
## [7453] 5 6 6 5 5 4 5 6 5 4 5 5 4 4 3 4 5 6 5 4 5 4 3
## [7476] 4 5 6 7 6 5 6 6 5 5 4 5 6 5 4 5 5 4 4 3 4 5 6
## [7499] 7 6 5 6 5 4 5 6 5 4 5 4 3 4 5 6 5 4 5 5 4 4 3
## [7522] 4 5 4 3 4 4 3 3 2 3 4 5 6 7 8 7 6 7 7 6 6 5 6
## [7545] 7 6 5 6 6 5 5 4 5 6 7 8 7 6 7 6 5 6 7 6 5 6 5
## [7568] 4 5 6 7 6 5 6 6 5 5 4 5 6 5 4 5 5 4 4 3 4 5 6
## [7591] 5 4 5 4 3 4 5 6 7 6 5 6 6 5 5 4 5 6 5 4 5 5 4
## [7614] 4 3 4 5 6 7 6 5 6 5 4 5 6 5 4 5 4 3 4 5 6 5 4
## [7637] 5 5 4 4 3 4 5 4 3 4 4 3 3 2 3 4 5 6 7 6 5 6 6
## [7660] 5 5 4 5 6 5 4 5 5 4 4 3 4 5 6 7 6 5 6 5 4 5 6
## [7683] 5 4 5 4 3 4 5 6 5 4 5 5 4 4 3 4 5 4 3 4 4 3 3
## [7706] 2 3 4 5 4 3 4 3 2 3 4 5 6 5 4 5 5 4 4 3 4 5 4
## [7729] 3 4 4 3 3 2 3 4 5 6 5 4 5 4 3 4 5 4 3 4 3 2 3
## [7752] 4 5 4 3 4 4 3 3 2 3 4 3 2 3 3 2 2largest_cliques(g11) # cliques with max number of nodes## [[1]]
## + 11/65 vertices, named, from 861146a:
## [1] Harry Potter Hermione Granger Ron Weasley Albus Dumbledore
## [5] Arthur Weasley Molly Weasley Fred Weasley George Weasley
## [9] Ginny Weasley Bill Weasley Charlie Weasley
##
## [[2]]
## + 11/65 vertices, named, from 861146a:
## [1] Harry Potter Hermione Granger Ron Weasley Albus Dumbledore
## [5] Lord Voldemort Charlie Weasley Bill Weasley Fred Weasley
## [9] George Weasley Ginny Weasley Molly WeasleyFrom this plot, we can see that the biggest clique makes sense because it is composed of the Weasley family members as well as Ron Weasley’s best friends (Harry and Hermione) and their worst enemy (Voldemort). This implies, that without any of these characters, there wouldn’t be a story line. More importantly, the clique would probably break if Ron died because he’s the most important link in the clique (between the friends and the family).
#layout of the RMarkdown
par(mar=c(1,1,3,1))
#identifying the nodes within the network which belong to the clique
vcol <- rep("grey80", vcount(g11))
vcol[unlist(largest_cliques(g11))] <- "gold"
plot(g11,
layout = mylayout,
vertex.label = V(g11)$name,
vertex.color=vcol,
vertex.label.cex = 0.6,
vertex.label.dist=6,
vertex.size = 25,
asp = 0.75,
rescale = FALSE,
ylim = c(-5,4.5),
xlim = c(-2,2))
legend("top", legend = "", cex = 1.0, bty = "n", ncol = 1,
title = "Harry Potter - Largest Clique")
We can also analyze dyads (pairs of two nodes), triads (groups of three nodes) and bigger cliques in our network. For dyads, we can use the function dyad_census() from igraph or dyad.census() from sna. Both are identical and calculate a Holland and Leinhardt dyad census.
mut: 330 pairs with mutual connections. asym: 0 pairs with non-mutual connections (in the undirected network, there are none). null: 1750 pairs with no connection between them.
Thus, from our network, there are an additional 1750 pairs of relationships possible within the nodes. This goes to support that only certain characters have high degree centralities that connect to other characters.
adjacency <- as.matrix(as_adjacency_matrix(g11))
sna::dyad.census(adjacency)## Mut Asym Null
## [1,] 330 0 1750“A (maximal) clique is a maximal set of mutually adjacency vertices.”
In this case, we wanted an overall view on the rest of the cliques. In order words, we wanted to see the frequence distribution of the cliques based on the number of edges.
node_clique <- clique.census(adjacency, mode = "graph", tabulate.by.vertex = TRUE, clique.comembership = "sum")
edge_clique <- clique.census(adjacency, mode = "graph", tabulate.by.vertex = FALSE, clique.comembership = "sum")
edge_clique$clique.count## 1 2 3 4 5 6 7 8 9 10 11
## 0 13 12 14 22 9 13 2 2 6 2Community Detection
Clusters (Community detection based on edge betweenness (Newman-Girvan))
The Girvan-Newman algorithm detects communities by progressively removing edges from the original network. The connected components of the remaining network are the communities. Instead of trying to construct a measure that tells us which edges are the most central to communities, the Girvan-Newman algorithm focuses on edges that are most likely “between” communities.
From this network, the algorithm creates 13 communities.
#this is a list of all the cluster groups within the network
ceb <- cluster_edge_betweenness(g11)
ceb## IGRAPH clustering edge betweenness, groups: 13, mod: 0.041
## + groups:
## $`1`
## [1] "Regulus Arcturus Black" "Sirius Black"
## [3] "Lavender Brown" "Cho Chang"
## [5] "Vincent Crabbe Sr." "Vincent Crabbe"
## [7] "Bartemius Crouch Jr." "Fleur Delacour"
## [9] "Aberforth Dumbledore" "Albus Dumbledore"
## [11] "Argus Filch" "Seamus Finnigan"
## [13] "Cornelius Fudge" "Gregory Goyle"
## [15] "Hermione Granger" "Rubeus Hagrid"
## [17] "Bellatrix Lestrange" "Neville Longbottom"
## + ... omitted several groups/verticeslength(ceb) #number of communities## [1] 13membership(ceb) #community membership for each node - verify that each node was attributed a community## Regulus Arcturus Black Sirius Black Lavender Brown
## 1 1 1
## Cho Chang Vincent Crabbe Sr. Vincent Crabbe
## 1 1 1
## Bartemius Crouch Sr. Bartemius Crouch Jr. Fleur Delacour
## 2 1 1
## Cedric Diggory Aberforth Dumbledore Albus Dumbledore
## 3 1 1
## Dudley Dursley Petunia Dursley Vernon Dursley
## 4 4 4
## Argus Filch Seamus Finnigan Nicolas Flamel
## 1 1 5
## Cornelius Fudge Goyle Sr. Gregory Goyle
## 1 6 1
## Hermione Granger Rubeus Hagrid Igor Karkaroff
## 1 1 7
## Viktor Krum Bellatrix Lestrange Alice Longbottom
## 3 1 8
## Frank Longbottom Neville Longbottom Luna Lovegood
## 8 1 1
## Xenophilius Lovegood Remus Lupin Draco Malfoy
## 9 1 1
## Lucius Malfoy Narcissa Malfoy Olympe Maxime
## 1 1 7
## Minerva McGonagall Alastor Moody Peter Pettigrew
## 1 1 1
## Harry Potter James Potter Lily Potter
## 1 1 1
## Quirinus Quirrell Tom Riddle Sr. Mary Riddle
## 1 10 10
## Lord Voldemort Rita Skeeter Severus Snape
## 1 11 1
## Nymphadora Tonks Dolores Janes Umbridge Arthur Weasley
## 1 1 1
## Bill Weasley Charlie Weasley Fred Weasley
## 1 1 1
## George Weasley Ginny Weasley Molly Weasley
## 1 1 1
## Percy Weasley Ron Weasley Dobby
## 1 1 1
## Fluffy Hedwig Moaning Myrtle
## 1 12 13
## Aragog Grawp
## 1 1High modularity for a partitioning reflects dense connections within communities and sparse connections across communities. with this first algorithm, we calculated a low modularity of 0.04 because it creates the communites only based on the betweenness of the nodes.
modularity(ceb) # score of the graph partitioning ## [1] 0.04138659When plotting the clusters, we obtain:
#layout of the RMarkdown
par(mar=c(1,1,3,1))
plot(ceb,
g11,
layout = mylayout,
vertex.label = NA)
Community detection based on greedy optimization of modularity
After calculating a very low modularity with the cmmunity detection based on edge betweenness, we decided to create another community model based on greedy optimization, which is also the optimization of the modularity itself.
The aim of modularity-optimizing community detection algorithms is to determine the partition(s) with maximum modularity. However, because we are looking for a maximum over all partitions of a network, the number of which is exponential in N, maximising modularity is equivalent to comparing the modularity of each and every partition of the network to determine the best one.
From this algorithm, we were able to increase our modularity score to 0.23
When looking at our restult, it is a greedy community analysis algorithm that optimises the modularity score. This method starts with a totally non-clustered initial assignment, where each node forms a singleton community, and then computes the expected improvement of modularity for each pair of communities, chooses a community pair that gives the maximum improvement of modularity and merges them into a new community. The above procedure is repeated until no community pairs merge leads to an increase in modularity“.
cfg <- cluster_fast_greedy(g11)
modularity(cfg)## [1] 0.2334389When plotting the clusters, we obtain:
par(mar=c(0,0,0,0))
V(g11)$community <- cfg$membership
colrs <- adjustcolor( c("gray50", "tomato", "gold", "yellowgreen"), alpha=.6)
plot(g11,
layout = mylayout,
vertex.label = V(g11)$name,
vertex.color=colrs[V(g11)$community],
vertex.label.color = "black",
vertex.label.cex = 0.6,
vertex.label.dist=3,
vertex.size = 20,
legend = TRUE,
asp = 0.75,
rescale = FALSE,
ylim = c(-4.5,5),
xlim = c(-3,2))
legend("top", legend = "", cex = 2, bty = "n", ncol = 1,
title = "Harry Potter - Communities")
When wanting to have a more descriptive view of the dataset, we created a data frame to see which nodes belong to which community.
fg_HP<-fastgreedy.community(igraph::simplify(g1,edge.attr.comb = "min"))
keys_community<-keys
keys_community$grouping<-as.factor(membership(fg_HP))
keys_community$Gender_Male<-NULL
keys_community$Color<-NULL
head(keys_community[order(-keys_community$grouping),])## Warning in Ops.factor(keys_community$grouping): '-' not meaningful for
## factors## Character Alignment House grouping
## 1 Regulus Arcturus Black Good Slytherin 1
## 2 Sirius Black Good Gryffindor 2
## 3 Lavender Brown Good Gryffindor 4
## 4 Cho Chang Good Ravenclaw 4
## 5 Vincent Crabbe Sr. Evil Slytherin 1
## 6 Vincent Crabbe Evil Slytherin 1Next, the following graph is just eye candy. It is a fun way to replicate the above graph in a more interactive way. The graph gives you the possibility to move the nodes around and with the filter on the left, select particular nodes (characters) and their closest relationships with other nodes.
(The graph is interactive, meaning you can alo zoom in/out but also can click on specific nodes and move them around. Plus, if the graph does not show in the Markdown, make sure it is in the same zip folder that the R document)
#Sets node information based on actors as well as their associated community
nodes <- data.frame(id = V(g11)$name, title = V(g11)$name, group = V(g11)$community)
nodes <- nodes[order(nodes$id, decreasing = F),]
edges <- get.data.frame(g11, what="edges")[1:2]
graphh<-visNetwork(nodes, edges) %>%
visOptions(highlightNearest = TRUE, nodesIdSelection = TRUE) %>%
visLegend(position = "right", main = "Group")
visPhysics(graphh, stabilization = FALSE, adaptiveTimestep = FALSE, timestep = FALSE)Network Visualization
As for the network visualization part, this will be explored both in Gephi and in Netlogo.
Conclusion
In conclusion, our analysis confirms Harry Potter as the most important character in the network. However, depending on your objective, it might be better to choose other characters that may have higher levels of degree or better diffusion capabilities. For example: Hermione seems to have the most influence on the whole network.
In all, if you need to pick a character to play in the future Harry Potter game, choose a character that belongs to the Gryffindor house and is aligned as a Good character. The chances of success may be greater, as most metrics identify these categories of characters as the most “connected”.